1799:
8333:
1487:
8642:
1794:{\displaystyle {\begin{aligned}G(r,c)&=\sum _{k=0}^{\infty }cr^{k}&&\\&=c+\sum _{k=0}^{\infty }cr^{k+1}&&{\text{ (stability) }}\\&=c+r\sum _{k=0}^{\infty }cr^{k}&&{\text{ (linearity) }}\\&=c+r\,G(r,c),&&{\text{ hence }}\\G(r,c)&={\frac {c}{1-r}},{\text{ unless it is infinite}}&&\\\end{aligned}}}
388:
in 1880; Cesàro's key contribution was not the discovery of this method, but his idea that one should give an explicit definition of the sum of a divergent series.) In the years after Cesàro's paper, several other mathematicians gave other definitions of the sum of a divergent series, although these
43:
Les séries divergentes sont en général quelque chose de bien fatal et c’est une honte qu’on ose y fonder aucune démonstration. ("Divergent series are in general something fatal, and it is a disgrace to base any proof on them." Often translated as "Divergent series are an invention of the devil …")
868:
Summation methods usually concentrate on the sequence of partial sums of the series. While this sequence does not converge, we may often find that when we take an average of larger and larger numbers of initial terms of the sequence, the average converges, and we can use this average instead of a
1817:
The two classical summation methods for series, ordinary convergence and absolute convergence, define the sum as a limit of certain partial sums. These are included only for completeness; strictly speaking they are not true summation methods for divergent series since, by definition, a series is
7310:
7638:
5212:
3527:
212:
7033:
3941:
4986:
88:
If a series converges, the individual terms of the series must approach zero. Thus any series in which the individual terms do not approach zero diverges. However, convergence is a stronger condition: not all series whose terms approach zero converge. A
6601:
4791:
4154:
557:
7167:
2169:
5794:
4482:
4341:
6452:
6875:
1467:
Taking regularity, linearity and stability as axioms, it is possible to sum many divergent series by elementary algebraic manipulations. This partly explains why many different summation methods give the same answer for certain series.
389:
are not always compatible: different definitions can give different answers for the sum of the same divergent series; so, when talking about the sum of a divergent series, it is necessary to specify which summation method one is using.
6712:
6281:
1854:, if it exists. It does not depend on the order of the elements of the sequence, and a classical theorem says that a sequence is absolutely convergent if and only if the sequence of absolute values is convergent in the standard sense.
7867:. Muraev observes that Borel summation is translative in one of the two directions: augmenting a series by a zero placed at its start does not change the summability or value of the series. However, he states "the converse is false".
5453:
2831:
5820:. Since the hyperreal numbers include distinct infinite values, these numbers can be used to represent the values of divergent series. The key method is to designate a particular infinite value that is being summed, usually
367:
and others, but often led to confusing and contradictory results. A major problem was Euler's idea that any divergent series should have a natural sum, without first defining what is meant by the sum of a divergent series.
2984:
7193:
5666:
7779:
7337:
2001:
3289: = 1. This value may depend on the choice of path. One of the first examples of potentially different sums for a divergent series, using analytic continuation, was given by Callet, who observed that if
884:
that assigns the same values to the corresponding series. There are certain properties it is desirable for these methods to possess if they are to arrive at values corresponding to limits and sums, respectively.
1804:
can be evaluated regardless of convergence. More rigorously, any summation method that possesses these properties and which assigns a finite value to the geometric series must assign this value. However, when
2535:
5060:
2634:
6023:
3155:
1254:
3740:
3326:
1492:
228:
In specialized mathematical contexts, values can be objectively assigned to certain series whose sequences of partial sums diverge, in order to make meaning of the divergence of the series. A
102:
703:
664:
372:
eventually gave a rigorous definition of the sum of a (convergent) series, and for some time after this, divergent series were mostly excluded from mathematics. They reappeared in 1886 with
3617:
593:
475:
466:
430:
2299:
6901:
3768:
4852:
5320:
5931:
6057:
8524:
8142:
Alexander I. Saichev and Wojbor
Woyczynski:"Distributions in the Physical and Engineering Sciences, Volume 1", Chap.8 "Summation of divergent series and integrals", Springer (2018).
7902:
790:. This fact is not very useful in practice, since there are many such extensions, inconsistent with each other, and also since proving such operators exist requires invoking the
6089:
In 1812 Hutton introduced a method of summing divergent series by starting with the sequence of partial sums, and repeatedly applying the operation of replacing a sequence
625:
6490:
4644:
3993:
1312:
5510:
292:
5976:
7064:
3319:
2016:
470:
4212:
The operation of Euler summation can be repeated several times, and this is essentially equivalent to taking an analytic continuation of a power series to the point
7670:
5692:
4377:
4230:
5878:
5858:
5838:
3985:
3961:
3657:
8514:
6330:
6777:
8167:
6628:
6184:
3555:
5345:
3760:
8607:
2710:
1818:
divergent only if these methods do not work. Most but not all summation methods for divergent series extend these methods to a larger class of sequences.
8448:
353:
1 − 1 + 1...?', and that this habit of mind led them into unnecessary perplexities and controversies which were often really verbal.
2890:
7305:{\displaystyle \lim _{\omega \rightarrow \infty }{\frac {\kappa }{\omega ^{\kappa }}}\int _{0}^{\omega }A_{\lambda }(x)(\omega -x)^{\kappa -1}\,dx.}
7633:{\displaystyle \lim _{m\rightarrow \infty }\sum _{k=0}^{m}a_{k}{\frac {^{2}}{\Gamma (m+1-k)\,\Gamma (m+1+k)}}=\lim _{m\rightarrow \infty }\left=s,}
4209:. Euler used it before analytic continuation was defined in general, and gave explicit formulas for the power series of the analytic continuation.
5579:
8458:
7692:
3184:
goes to positive zero. The Lindelöf sum is a powerful method when applied to power series among other applications, summing power series in the
6729:
1925:
8453:
5207:{\displaystyle \lim _{n\rightarrow +\infty }{\sqrt {\frac {H(n)}{2\pi }}}\sum _{h\in Z}e^{-{\frac {1}{2}}h^{2}H(n)}(a_{0}+\cdots +a_{h})}
5799:
if this integral exists. A further generalization is to replace the sum under the integral by its analytic continuation from small
3988:
2454:
8622:
8213:
8160:
2562:
8602:
8504:
8494:
673:
561:
434:
5981:
3051:
634:
566:
439:
403:
8612:
6465:
then it is
Lambert summable to the same value, and if a series is Lambert summable then it is Abel summable to the same value.
7790:
3522:{\displaystyle {\frac {1-x^{m}}{1-x^{n}}}={\frac {1+x+\dots +x^{m-1}}{1+x+\dots x^{n-1}}}=1-x^{m}+x^{n}-x^{n+m}+x^{2n}-\dots }
1223:
3664:
1809:
is a real number larger than 1, the partial sums increase without bound, and averaging methods assign a limit of infinity.
8617:
8519:
8153:
8129:
8111:
8092:
3964:
818:
218:
207:{\displaystyle 1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}+\cdots =\sum _{n=1}^{\infty }{\frac {1}{n}}.}
94:
8136:
Werner Balser: "From
Divergent Power Series to Analytic Functions", Springer-Verlag, LNM 1582, ISBN 0-387-58268-1 (1994).
4169:
Euler summation is essentially an explicit form of analytic continuation. If a power series converges for small complex
3017:
whenever the latter is defined. The Abel sum is therefore regular, linear, stable, and consistent with Cesàro summation.
8681:
8676:
8671:
8645:
4513:
4366:
1453:
845:
8686:
8627:
8124:
8106:
8087:
1826:
Absolute convergence defines the sum of a sequence (or set) of numbers to be the limit of the net of all partial sums
602:
7916:
Bartlett, Jonathan; Gaastra, Logan; Nemati, David (January 2020). "Hyperreal
Numbers for Infinite Divergent Series".
7028:{\displaystyle \lim _{h\rightarrow 0}{\frac {2}{\pi }}\sum _{n}{\frac {\sin ^{2}nh}{n^{2}h}}(a_{1}+\cdots +a_{n})=s.}
3936:{\displaystyle \sum _{k\geq 0}(-1)^{k+1}{\frac {1}{2k-1}}{\binom {2k}{k}}=1+2-2+4-10+28-84+264-858+2860-9724+\cdots }
8499:
8489:
8479:
4981:{\displaystyle \lim _{x\rightarrow \infty }{\frac {\sum _{n}p_{n}(a_{0}+\cdots +a_{n})x^{n}}{\sum _{n}p_{n}x^{n}}},}
4357: = 0 is an isolated singularity, the sum is defined by the constant term of the Laurent series expansion.
3562:
668:
629:
597:
8509:
4350: = 0, if this exists and is unique. This method is sometimes confused with zeta function regularization.
2660:
2225:
1479:
8139:
William O. Bray and Časlav V. Stanojević(Eds.): "Analysis of
Divergence", Springer, ISBN 978-1-4612-7467-4 (1999).
7828:
8666:
385:
5275:
8082:
7681:
5886:
5840:, which is used as a unit of infinity. Instead of summing to an arbitrary infinity (as is typically done with
784:
that it may be extended to a summation method summing any series with bounded partial sums. This is called the
332:
6028:
4601:
781:
4224:
This method defines the sum of a series to be the value of the analytic continuation of the
Dirichlet series
8594:
8416:
8119:
7903:"The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation"
6728:
Ramanujan summation is a method of assigning a value to divergent series used by
Ramanujan and based on the
5937:
2667:
Abelian means are regular and linear, but not stable and not always consistent between different choices of
1449:
837:
31:
8101:
1448:
There are powerful numerical summation methods that are neither regular nor linear, for instance nonlinear
1441:.) If two methods are consistent, and one sums more series than the other, the one summing more series is
331:, there are a wide variety of summability methods; these are discussed in greater detail in the article on
8256:
8203:
6596:{\displaystyle \lim _{\zeta \rightarrow 1^{-}}\sum _{n}{\frac {\Gamma (1+\zeta n)}{\Gamma (1+n)}}a_{n}=s.}
4786:{\displaystyle \zeta (-s)=\sum _{n=1}^{\infty }n^{s}=1^{s}+2^{s}+3^{s}+\cdots =-{\frac {B_{s+1}}{s+1}}\,,}
4149:{\displaystyle \ldots =\sum _{k\geq 0}(-4)^{k}{\frac {(-1/2)_{k}}{k!}}={}_{1}F_{0}(-1/2;;-4)={\sqrt {5}}.}
552:{\displaystyle {\text{ “ = ” }}\!\!\int _{0}^{\infty }{\frac {e^{-x}}{1+x}}\,dx\approx 0.596\,347\ldots }
8463:
8208:
8043:
4568:
4501:
3250:
877:
is any summation method assigning values to a set of sequences, we may mechanically translate this to a
802:
369:
324:
2187:
1289:
5476:
8574:
8411:
8180:
7162:{\displaystyle A_{\lambda }(x)=a_{0}+\cdots +a_{n}{\text{ for }}\lambda _{n}<x\leq \lambda _{n+1}}
2164:{\displaystyle t_{m}={\frac {p_{m}s_{0}+p_{m-1}s_{1}+\cdots +p_{0}s_{m}}{p_{0}+p_{1}+\cdots +p_{m}}}}
253:
35:
5054:. Valiron showed that under certain conditions it is equivalent to defining the sum of a series as
380:
realized that one could give a rigorous definition of the sum of some divergent series, and defined
8554:
8421:
6723:
6079:
5943:
5050:
Valiron's method is a generalization of Borel summation to certain more general integral functions
3185:
2990:
2370:
1457:
853:
841:
810:
381:
241:
82:
5789:{\displaystyle \int _{0}^{\infty }e^{-t}\sum {\frac {a_{n}t^{n\alpha }}{\Gamma (n\alpha +1)}}\,dt}
3292:
8395:
8380:
8352:
8332:
8271:
7943:
7939:
7925:
4477:{\displaystyle f(s)={\frac {1}{a_{1}^{s}}}+{\frac {1}{a_{2}^{s}}}+{\frac {1}{a_{3}^{s}}}+\cdots }
4336:{\displaystyle f(s)={\frac {a_{1}}{1^{s}}}+{\frac {a_{2}}{2^{s}}}+{\frac {a_{3}}{3^{s}}}+\cdots }
2207:
The Nørlund mean is regular, linear, and stable. Moreover, any two Nørlund means are consistent.
1064:′ is the sequence obtained by omitting the first value and subtracting it from the rest, so that
743:
8484:
7646:
398:
245:
6447:{\displaystyle \lim _{y\rightarrow 0^{+}}\sum _{n\geq 1}a_{n}{\frac {nye^{-ny}}{1-e^{-ny}}}=s.}
2215:
The most significant of the Nørlund means are the Cesàro sums. Here, if we define the sequence
8584:
8385:
8357:
8311:
8301:
8281:
8266:
5817:
2680:
857:
771:
only summed convergent series (making it useless as a summation method for divergent series).
738:
724:
377:
373:
70:
6870:{\displaystyle \lim _{h\rightarrow 0}\sum _{n}a_{n}\left({\frac {\sin nh}{nh}}\right)^{k}=s.}
6748:
at non-integral points, so it is not really a summation method in the sense of this article.
5863:
5843:
5823:
3970:
3946:
3624:
8569:
8316:
8306:
8286:
8188:
7935:
6707:{\displaystyle \lim _{\delta \rightarrow 0}\sum _{n}{\frac {a_{n}}{\Gamma (1+\delta n)}}=s.}
6317:
6276:{\displaystyle \lim _{x\rightarrow \infty }\sum _{1\leq n\leq x}a_{n}{\frac {n}{x}}\left=s.}
4804:
3200:
2655:
826:
345:... but it is broadly true to say that mathematicians before Cauchy asked not 'How shall we
237:
7863:
2989:
Abel summation is interesting in part because it is consistent with but more powerful than
767:
was convergent in the first place; without any side-condition such a result would say that
8347:
8276:
7859:
5570:
5448:{\displaystyle a(x)={\frac {a_{0}x^{0}}{\mu _{0}}}+{\frac {a_{1}x^{1}}{\mu _{1}}}+\cdots }
5039:
4164:
1461:
1196:
849:
814:
791:
729:
320:
66:
4530: + ... Zeta function regularization is nonlinear. In applications, the numbers
3534:
795:
8579:
8564:
8559:
8238:
8223:
8060:
3745:
2857:
2826:{\displaystyle f(x)=\sum _{n=0}^{\infty }a_{n}e^{-nx}=\sum _{n=0}^{\infty }a_{n}z^{n},}
822:
806:
364:
90:
8660:
8544:
8218:
7947:
6297:) (Cesàro) summability implies Ingham summability, and Ingham summability implies (C,
6286:
5940:
in an infinite context. For instance, using this method, the sum of the progression
833:
748:
222:
8549:
8291:
8233:
5816:
This summation method works by using an extension to the real numbers known as the
2861:
786:
52:, letter to Holmboe, January 1826, reprinted in volume 2 of his collected papers.
8296:
8243:
8056:
2651:, then one can still define the sum of the divergent series by the limit above.
873:
can be seen as a function from a set of sequences of partial sums to values. If
384:. (This was not the first use of Cesàro summation, which was used implicitly by
78:
58:
2979:{\displaystyle A(s)=\lim _{z\rightarrow 1^{-}}\sum _{n=0}^{\infty }a_{n}z^{n}.}
7044:
1407:
1370:, with only finitely many terms re-indexed.) This is a weaker condition than
1188:
49:
3765:
Another example of analytic continuation is the divergent alternating series
3285: = 1, then the sum of the series can be defined to be the value at
1195:
The third condition is less important, and some significant methods, such as
8228:
7810:
1218:
315:
5661:{\displaystyle \int _{0}^{\infty }e^{-t}\sum {\frac {a_{n}t^{n}}{n!}}\,dt.}
947:
if it is a linear functional on the sequences where it is defined, so that
17:
8024:
Large-Order
Perturbation Theory and Summation Methods in Quantum Mechanics
7774:{\displaystyle \lim _{\alpha \to 0^{+}}\sum _{n}c_{n}e^{-\alpha n^{2}}=s.}
1460:, as well as the order-dependent mappings of perturbative series based on
8176:
74:
5860:), the BGN method sums to the specific hyperreal infinite value labeled
5010:. In this case one defines the sum as above, except taking the limit as
1996:{\displaystyle {\frac {p_{n}}{p_{0}+p_{1}+\cdots +p_{n}}}\rightarrow 0.}
8145:
2438:} is a strictly increasing sequence tending towards infinity, and that
328:
4173:
and can be analytically continued to the open disk with diameter from
821:
marked an epoch in the subject, introducing unexpected connections to
805:, is primarily concerned with explicit and natural techniques such as
5936:
This allows the usage of standard formulas for finite series such as
5524:
increase too rapidly then they do not uniquely determine the measure
2671:. However, some special cases are very important summation methods.
7930:
919:
Equivalently, the corresponding series-summation method evaluates
2876:
approaches 1 from below through positive reals, and the Abel sum
2530:{\displaystyle f(x)=\sum _{n=0}^{\infty }a_{n}e^{-\lambda _{n}x}}
1879:
defines the sum to be the limit of the sequence of partial sums
1202:
One can also give a weaker alternative to the last condition.
8149:
6025:, or, using just the most significant infinite hyperreal part,
5470:) sum of the series is defined to be the value of the integral
2629:{\displaystyle A_{\lambda }(s)=\lim _{x\rightarrow 0^{+}}f(x).}
1897:. This is the default definition of convergence of a sequence.
7971:
7969:
7850:-multiple series, and entire functions associated with them",
7327: + ... is called VP (or Vallée-Poussin) summable to
3742:, so different sums correspond to different placements of the
1191:
must be valid for the series that are summable by this method.
363:
Before the 19th century, divergent series were widely used by
6018:{\displaystyle {\frac {\omega ^{2}}{2}}+{\frac {\omega }{2}}}
3150:{\displaystyle f(x)=a_{1}+a_{2}2^{-2x}+a_{3}3^{-3x}+\cdots .}
2658:; in applications to physics, this is known as the method of
6618: + ... is called Mittag-Leffler (M) summable to
5671:
There is a generalization of this depending on a variable
5269:
is a measure on the real line such that all the moments
4994:
There is a variation of this method where the series for
4609:
can also be used to assign values for the divergent sums
4539:
are sometimes the eigenvalues of a self-adjoint operator
3249:
Several summation methods involve taking the value of an
2355:, and hence are regular, linear, stable, and consistent.
840:
as numerical techniques. Examples of such techniques are
8001:
7999:
7986:
7984:
1386:
A desirable property for two distinct summation methods
3277:
and can be analytically continued along some path from
2647:
but can be analytically continued to all positive real
323:
of the sequence of partial sums. Other methods involve
3233:
in the Mittag-Leffler star. Moreover, convergence to
1437:
is regular iff it is consistent with the standard sum
1249:{\displaystyle f:\mathbb {N} \rightarrow \mathbb {N} }
774:
The function giving the sum of a convergent series is
8525:
1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)
8515:
1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)
7695:
7649:
7340:
7196:
7067:
6904:
6780:
6631:
6493:
6333:
6187:
6031:
5984:
5946:
5889:
5866:
5846:
5826:
5695:
5582:
5479:
5348:
5278:
5063:
4855:
4647:
4380:
4233:
3996:
3973:
3949:
3771:
3748:
3735:{\displaystyle 1-1+0+1-1+0+1-1+\dots ={\frac {1}{3}}}
3667:
3627:
3565:
3537:
3329:
3295:
3199:) is analytic in a disk around zero, and hence has a
3054:
2893:
2713:
2565:
2457:
2228:
2019:
1928:
1862:
Cauchy's classical definition of the sum of a series
1490:
1292:
1226:
1039:
being a linear functional on the terms of the series.
676:
637:
605:
569:
478:
442:
406:
256:
105:
8022:
Arteca, G.A.; Fernández, F.M.; Castro, E.A. (1990),
776:
741:. More subtle, are partial converse results, called
719:
8593:
8537:
8472:
8441:
8434:
8404:
8373:
8366:
8340:
8252:
8196:
8187:
7773:
7664:
7632:
7304:
7161:
7027:
6869:
6706:
6595:
6446:
6275:
6051:
6017:
5970:
5925:
5872:
5852:
5832:
5788:
5660:
5504:
5447:
5314:
5206:
4980:
4785:
4476:
4335:
4148:
3979:
3955:
3935:
3754:
3734:
3651:
3611:
3549:
3521:
3313:
3149:
2978:
2825:
2628:
2529:
2293:
2163:
1995:
1793:
1306:
1248:
697:
658:
619:
587:
551:
460:
424:
286:
206:
3855:
3837:
2285:
2250:
832:Summation of divergent series is also related to
485:
484:
7697:
7480:
7342:
7198:
7058:form an increasing sequence of real numbers and
6906:
6782:
6744:at integers, but also on values of the function
6633:
6495:
6335:
6189:
5065:
4857:
2910:
2654:A series of this type is known as a generalized
2589:
801:The subject of divergent series, as a domain of
709:Theorems on methods for summing divergent series
698:{\displaystyle {\text{ “ = ” }}-{\frac {1}{12}}}
4192:to 1 and is continuous at 1, then its value at
1912:is a sequence of positive terms, starting from
659:{\displaystyle {\text{ “ = ” }}-{\frac {1}{2}}}
343:
240:from the set of series to values. For example,
41:
6480: + ... is called Le Roy summable to
6174: + ... is called Ingham summable to
3967:symbols. Using the duplication formula of the
3612:{\displaystyle 1-1+1-1+\dots ={\frac {m}{n}}.}
3210:) with a positive radius of convergence, then
588:{\displaystyle {\text{ “ = ” }}{\frac {1}{3}}}
461:{\displaystyle {\text{ “ = ” }}{\frac {1}{4}}}
425:{\displaystyle {\text{ “ = ” }}{\frac {1}{2}}}
8161:
8032:Baker, Jr., G. A.; Graves-Morris, P. (1996),
5880:. Therefore, the summations are of the form
3621:However, the gaps in the series are key. For
3241:) is uniform on compact subsets of the star.
2294:{\displaystyle p_{n}^{k}={n+k-1 \choose k-1}}
1374:, because any summation method that exhibits
349:1 − 1 + 1...?' but 'What
8:
8608:Hypergeometric function of a matrix argument
8073:Large-Order Behaviour of Perturbation Theory
7940:10.33014/issn.2640-5652.2.1.bartlett-et-al.1
6740:(1) + ... depends not only on the values of
4599:, assigning a value to the divergent series
8464:1 + 1/2 + 1/3 + ... (Riemann zeta function)
869:limit to evaluate the sum of the series. A
8438:
8370:
8193:
8168:
8154:
8146:
8071:LeGuillou, J.-C.; Zinn-Justin, J. (1990),
8048:Extrapolation Methods. Theory and Practice
7846:Muraev, E. B. (1978), "Borel summation of
5315:{\displaystyle \mu _{n}=\int x^{n}\,d\mu }
2185:goes to infinity is an average called the
2006:If now we transform a sequence s by using
1217:′ are two series such that there exists a
848:, and order-dependent mappings related to
723:if it agrees with the actual limit on all
8520:1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)
7929:
7754:
7743:
7733:
7723:
7711:
7700:
7694:
7648:
7554:
7548:
7523:
7517:
7504:
7483:
7448:
7416:
7388:
7382:
7372:
7361:
7345:
7339:
7292:
7280:
7249:
7239:
7234:
7222:
7213:
7201:
7195:
7147:
7128:
7119:
7113:
7094:
7072:
7066:
7007:
6988:
6969:
6948:
6941:
6935:
6921:
6909:
6903:
6852:
6822:
6811:
6801:
6785:
6779:
6664:
6658:
6652:
6636:
6630:
6578:
6527:
6521:
6509:
6498:
6492:
6420:
6396:
6383:
6377:
6361:
6349:
6338:
6332:
6250:
6236:
6230:
6208:
6192:
6186:
6038:
6032:
6030:
6005:
5991:
5985:
5983:
5945:
5926:{\displaystyle \sum _{x=1}^{\omega }f(x)}
5905:
5894:
5888:
5865:
5845:
5825:
5779:
5744:
5734:
5727:
5715:
5705:
5700:
5694:
5648:
5631:
5621:
5614:
5602:
5592:
5587:
5581:
5495:
5478:
5431:
5420:
5410:
5403:
5392:
5381:
5371:
5364:
5347:
5305:
5299:
5283:
5277:
5195:
5176:
5149:
5135:
5131:
5115:
5083:
5068:
5062:
4966:
4956:
4946:
4934:
4921:
4902:
4889:
4879:
4872:
4860:
4854:
4779:
4754:
4748:
4730:
4717:
4704:
4691:
4681:
4670:
4646:
4460:
4455:
4446:
4435:
4430:
4421:
4410:
4405:
4396:
4379:
4319:
4309:
4303:
4292:
4282:
4276:
4265:
4255:
4249:
4232:
4220:Analytic continuation of Dirichlet series
4136:
4110:
4095:
4085:
4083:
4062:
4050:
4038:
4032:
4007:
3995:
3972:
3948:
3854:
3836:
3834:
3813:
3801:
3776:
3770:
3747:
3722:
3666:
3626:
3596:
3564:
3536:
3504:
3485:
3472:
3459:
3431:
3398:
3373:
3361:
3343:
3330:
3328:
3294:
3126:
3116:
3097:
3087:
3074:
3053:
2967:
2957:
2947:
2936:
2924:
2913:
2892:
2814:
2804:
2794:
2783:
2764:
2754:
2744:
2733:
2712:
2603:
2592:
2570:
2564:
2516:
2508:
2498:
2488:
2477:
2456:
2284:
2249:
2247:
2238:
2233:
2227:
2152:
2133:
2120:
2108:
2098:
2079:
2063:
2050:
2040:
2033:
2024:
2018:
1978:
1959:
1946:
1935:
1929:
1927:
1780:
1759:
1725:
1700:
1676:
1667:
1654:
1643:
1615:
1600:
1587:
1576:
1548:
1535:
1524:
1491:
1489:
1433:(Using this language, a summation method
1300:
1299:
1291:
1242:
1241:
1234:
1233:
1225:
817:, and their relationships. The advent of
685:
677:
675:
646:
638:
636:
606:
604:
575:
570:
568:
542:
529:
507:
501:
495:
490:
479:
477:
448:
443:
441:
412:
407:
405:
255:
191:
185:
174:
151:
138:
125:
112:
104:
6052:{\displaystyle {\frac {\omega ^{2}}{2}}}
5573:, where the value of a sum is given by
2373:. Cesàro sums have the property that if
1187:Another way of stating this is that the
7802:
6159:, ..., and then taking the limit.
6103:, ... by the sequence of averages
5339: + ... is a series such that
1031:are linear functionals on the sequence
376:'s work on asymptotic series. In 1890,
7876:
1035:and vice versa, this is equivalent to
798:. They are therefore nonconstructive.
763:, and some side-condition holds, then
217:The divergence of the harmonic series
27:Infinite series that is not convergent
8005:
7990:
7975:
7960:
7918:Communications of the Blyth Institute
7888:
6067:
3257:Analytic continuation of power series
358:G. H. Hardy, Divergent series, page 6
81:of the series does not have a finite
7:
7686:The series is Zeldovich summable if
5253:is to be interpreted as 0 when
8485:1 − 1 + 1 − 1 + ⋯ (Grandi's series)
4998:has a finite radius of convergence
4559:has eigenvalues 1, 2, 3, ... then
4508: = −1, then its value at
3659:for example, we actually would get
3045:, then (indexing from one) we have
1454:Levin-type sequence transformations
846:Levin-type sequence transformations
7650:
7490:
7449:
7424:
7394:
7352:
7208:
7187: + ... is defined to be
6671:
6553:
6530:
6199:
5847:
5755:
5706:
5686: + ... is defined to be
5593:
5515:if it is defined. (If the numbers
5078:
4867:
4846: + ... is defined to be
4835:is an integral function, then the
4682:
3974:
3950:
3841:
2948:
2795:
2745:
2639:More generally, if the series for
2489:
2254:
1655:
1588:
1536:
620:{\displaystyle {\text{ “ = ” }}-1}
496:
186:
25:
8603:Generalized hypergeometric series
6730:Euler–Maclaurin summation formula
5679:) sum, where the sum of a series
5569:!, and this gives one version of
3989:generalized hypergeometric series
2348:Cesàro sums are Nørlund means if
1307:{\displaystyle N\in \mathbb {N} }
319:method, in that it relies on the
8641:
8640:
8613:Lauricella hypergeometric series
8331:
6732:. The Ramanujan sum of a series
6293:is any positive number then (C,−
5505:{\displaystyle \int a(x)\,d\mu }
3943:which is a sum over products of
2010:to give weighted means, setting
1382:, but the converse is not true.)
8623:Riemann's differential equation
7833:The Encyclopedia of Mathematics
2700:, then we obtain the method of
2540:converges for all real numbers
864:Properties of summation methods
679: “ = ”
640: “ = ”
608: “ = ”
572: “ = ”
481: “ = ”
445: “ = ”
409: “ = ”
287:{\displaystyle 1-1+1-1+\cdots }
7901:Tao, Terence (10 April 2010).
7704:
7659:
7653:
7604:
7592:
7589:
7577:
7572:
7560:
7487:
7470:
7452:
7445:
7427:
7413:
7409:
7397:
7391:
7349:
7277:
7264:
7261:
7255:
7205:
7084:
7078:
7013:
6981:
6913:
6789:
6763: + ... is called (R,
6689:
6674:
6640:
6568:
6556:
6548:
6533:
6502:
6342:
6196:
5920:
5914:
5773:
5758:
5492:
5486:
5358:
5352:
5201:
5169:
5164:
5158:
5096:
5090:
5072:
5038:this gives one (weak) form of
4927:
4895:
4864:
4660:
4651:
4512: = −1 is called the
4390:
4384:
4243:
4237:
4130:
4101:
4059:
4041:
4029:
4019:
3798:
3788:
3064:
3058:
2917:
2903:
2897:
2723:
2717:
2620:
2614:
2596:
2582:
2576:
2467:
2461:
1987:
1749:
1737:
1716:
1704:
1510:
1498:
1414:to which both assign a value,
1238:
221:by the medieval mathematician
1:
8618:Modular hypergeometric series
8459:1/4 + 1/16 + 1/64 + 1/256 + ⋯
5971:{\displaystyle 1+2+3+\ldots }
5244: + ... +
992:and a real or complex scalar
747:, from a prototype proved by
727:. Such a result is called an
8036:, Cambridge University Press
6887: + ... is called R
6461:) (Cesàro) summable for any
5221:is the second derivative of
4543:with compact resolvent, and
4487:(for positive values of the
4367:Zeta function regularization
4361:Zeta function regularization
3314:{\displaystyle 1\leq m<n}
3281: = 0 to the point
3273:converges for small complex
2544: > 0. Then the
1101:) is defined if and only if
1053:is a sequence starting from
794:or its equivalents, such as
471:1 − 1 + 2 − 6 + 24 − 120 + ⋯
73:, meaning that the infinite
8628:Theta hypergeometric series
8125:Encyclopedia of Mathematics
8107:Encyclopedia of Mathematics
8088:Encyclopedia of Mathematics
8083:"Lindelöf summation method"
6767:) (or Riemann) summable to
6316: + ... is called
4496:) converges for large real
3987:-function, it reduces to a
2362:is ordinary summation, and
1813:Classical summation methods
1782: unless it is infinite
1286:, and if there exists some
852:techniques for large-order
34:. For the film series, see
8703:
8510:Infinite arithmetic series
8454:1/2 + 1/4 + 1/8 + 1/16 + ⋯
8449:1/2 − 1/4 + 1/8 − 1/16 + ⋯
8075:, Amsterdam: North-Holland
7791:Silverman–Toeplitz theorem
7679:
7665:{\displaystyle \Gamma (x)}
7316:Vallée-Poussin summability
7042:
6721:
6077:
4636:(−2) = 1 + 4 + 9 + ... = 0
4364:
4196:is called the Euler or (E,
4162:
2678:
2661:heat-kernel regularization
896:if, whenever the sequence
819:Wiener's tauberian theorem
780:, and it follows from the
29:
8636:
8329:
8065:, Oxford: Clarendon Press
8026:, Berlin: Springer-Verlag
6063:Hausdorff transformations
5026:In the special case when
4614:(0) = 1 + 1 + 1 + ... = −
2643:only converges for large
386:Ferdinand Georg Frobenius
313:. Cesàro summation is an
246:Grandi's divergent series
30:For the publication, see
8120:"Riesz summation method"
8100:Zakharov, A.A. (2001) ,
7682:Zeldovich regularization
6607:Mittag-Leffler summation
1450:sequence transformations
1366:′ is the same series as
892:. A summation method is
838:sequence transformations
737:, from the prototypical
8341:Properties of sequences
8102:"Abel summation method"
7672:is the gamma function.
5938:arithmetic progressions
5873:{\displaystyle \omega }
5853:{\displaystyle \infty }
5833:{\displaystyle \omega }
5812:BGN hyperreal summation
4811:Integral function means
4551:) is then the trace of
4504:along the real line to
3980:{\displaystyle \Gamma }
3956:{\displaystyle \Gamma }
3652:{\displaystyle m=1,n=3}
1678: (linearity)
1617: (stability)
1471:For instance, whenever
1135:Equivalently, whenever
879:series-summation method
32:Divergent (book series)
8204:Arithmetic progression
8081:Volkov, I.I. (2001) ,
7858:(6): 1332–1340, 1438,
7775:
7666:
7634:
7377:
7306:
7163:
7029:
6871:
6708:
6597:
6448:
6277:
6084:
6053:
6019:
5972:
5927:
5910:
5874:
5854:
5834:
5790:
5662:
5506:
5449:
5316:
5208:
5045:
5018:rather than infinity.
4991:if this limit exists.
4982:
4787:
4686:
4502:analytically continued
4478:
4337:
4150:
3981:
3957:
3937:
3756:
3736:
3653:
3613:
3551:
3523:
3315:
3151:
2980:
2952:
2827:
2799:
2749:
2630:
2531:
2493:
2295:
2165:
1997:
1795:
1659:
1592:
1540:
1410:if for every sequence
1380:finite re-indexability
1308:
1250:
1207:Finite re-indexability
699:
660:
621:
589:
553:
462:
426:
355:
327:of related series. In
325:analytic continuations
288:
208:
190:
46:
8595:Hypergeometric series
8209:Geometric progression
7776:
7676:Zeldovich summability
7667:
7635:
7357:
7307:
7164:
7030:
6872:
6709:
6598:
6449:
6278:
6054:
6020:
5973:
5928:
5890:
5875:
5855:
5835:
5807:Miscellaneous methods
5791:
5663:
5507:
5450:
5317:
5209:
4983:
4788:
4666:
4569:Riemann zeta function
4479:
4338:
4200:) sum of the series Σ
4151:
3982:
3958:
3938:
3757:
3737:
3654:
3614:
3552:
3524:
3316:
3251:analytic continuation
3245:Analytic continuation
3152:
2981:
2932:
2856:approaches 0 through
2844:). Then the limit of
2828:
2779:
2729:
2631:
2532:
2473:
2296:
2166:
1998:
1796:
1639:
1572:
1520:
1309:
1251:
1199:, do not possess it.
803:mathematical analysis
713:A summability method
700:
661:
622:
590:
554:
463:
427:
370:Augustin-Louis Cauchy
289:
209:
170:
8575:Trigonometric series
8367:Properties of series
8214:Harmonic progression
7693:
7647:
7338:
7194:
7180:) sum of the series
7065:
6902:
6778:
6629:
6491:
6331:
6185:
6029:
5982:
5944:
5887:
5864:
5844:
5824:
5693:
5580:
5477:
5346:
5276:
5061:
4853:
4645:
4583: = −1 is −
4378:
4231:
3994:
3971:
3947:
3769:
3746:
3665:
3625:
3563:
3535:
3327:
3293:
3052:
2891:
2860:is the limit of the
2711:
2563:
2455:
2304:then the Cesàro sum
2226:
2017:
1926:
1919:. Suppose also that
1822:Absolute convergence
1488:
1290:
1224:
674:
635:
603:
567:
476:
440:
404:
254:
103:
36:The Divergent Series
8682:Asymptotic analysis
8677:Summability methods
8672:Mathematical series
8555:Formal power series
7852:Akademiya Nauk SSSR
7815:Michon's Numericana
7811:"Summation methods"
7244:
6752:Riemann summability
6724:Ramanujan summation
6718:Ramanujan summation
6305:Lambert summability
5710:
5597:
5552:and 0 for negative
5257: < 0.
4602:1 + 2 + 3 + 4 + ...
4465:
4440:
4415:
3550:{\displaystyle x=1}
3186:Mittag-Leffler star
2243:
1109:′) is defined, and
854:perturbation theory
782:Hahn–Banach theorem
500:
8687:Summability theory
8353:Monotonic function
8272:Fibonacci sequence
7771:
7728:
7718:
7662:
7630:
7494:
7356:
7302:
7230:
7212:
7172:then the Riesz (R,
7159:
7025:
6940:
6920:
6867:
6806:
6796:
6704:
6657:
6647:
6593:
6526:
6516:
6457:If a series is (C,
6444:
6372:
6356:
6273:
6225:
6203:
6163:Ingham summability
6049:
6015:
5968:
5923:
5870:
5850:
5830:
5786:
5696:
5658:
5583:
5502:
5462:in the support of
5458:converges for all
5445:
5312:
5204:
5126:
5082:
4978:
4951:
4884:
4871:
4839:sum of the series
4783:
4605:. Other values of
4579:), whose value at
4555:. For example, if
4516:sum of the series
4474:
4451:
4426:
4401:
4333:
4146:
4018:
3977:
3953:
3933:
3787:
3752:
3732:
3649:
3609:
3547:
3519:
3311:
3172:, is the limit of
3147:
3021:Lindelöf summation
2976:
2931:
2840: = exp(−
2823:
2626:
2610:
2527:
2291:
2229:
2174:then the limit of
2161:
1993:
1791:
1789:
1304:
1246:
996:. Since the terms
744:Tauberian theorems
695:
656:
617:
585:
549:
486:
458:
422:
284:
230:summability method
204:
8654:
8653:
8585:Generating series
8533:
8532:
8505:1 − 2 + 4 − 8 + ⋯
8500:1 + 2 + 4 + 8 + ⋯
8495:1 − 2 + 3 − 4 + ⋯
8490:1 + 2 + 3 + 4 + ⋯
8480:1 + 1 + 1 + 1 + ⋯
8430:
8429:
8358:Periodic sequence
8327:
8326:
8312:Triangular number
8302:Pentagonal number
8282:Heptagonal number
8267:Complete sequence
8189:Integer sequences
8044:Redivo Zaglia, M.
8034:Padé Approximants
7719:
7696:
7608:
7539:
7479:
7474:
7341:
7228:
7197:
7122:
6979:
6931:
6929:
6905:
6846:
6797:
6781:
6693:
6648:
6632:
6572:
6517:
6494:
6433:
6357:
6334:
6258:
6244:
6204:
6188:
6047:
6013:
6000:
5818:hyperreal numbers
5777:
5675:, called the (B′,
5646:
5437:
5398:
5143:
5111:
5109:
5108:
5064:
4973:
4942:
4875:
4856:
4777:
4466:
4441:
4416:
4325:
4298:
4271:
4141:
4077:
4003:
3853:
3832:
3772:
3755:{\displaystyle 0}
3730:
3604:
3444:
3368:
2909:
2588:
2393:is stronger than
2283:
2159:
1985:
1783:
1775:
1728:
1727: hence
1679:
1618:
1458:Padé approximants
1362:(In other words,
858:quantum mechanics
842:Padé approximants
725:convergent series
693:
680:
669:1 + 2 + 3 + 4 + ⋯
654:
641:
630:1 + 1 + 1 + 1 + ⋯
609:
598:1 + 2 + 4 + 8 + ⋯
583:
573:
562:1 − 2 + 4 − 8 + ⋯
527:
482:
456:
446:
435:1 − 2 + 3 − 4 + ⋯
420:
410:
399:1 - 1 + 1 - 1 + ⋯
199:
159:
146:
133:
120:
16:(Redirected from
8694:
8667:Divergent series
8644:
8643:
8570:Dirichlet series
8439:
8371:
8335:
8307:Polygonal number
8287:Hexagonal number
8260:
8194:
8170:
8163:
8156:
8147:
8133:
8114:
8095:
8076:
8066:
8062:Divergent Series
8051:
8037:
8027:
8009:
8003:
7994:
7988:
7979:
7973:
7964:
7958:
7952:
7951:
7933:
7913:
7907:
7906:
7898:
7892:
7886:
7880:
7874:
7868:
7866:
7843:
7837:
7836:
7825:
7819:
7818:
7807:
7780:
7778:
7777:
7772:
7761:
7760:
7759:
7758:
7738:
7737:
7727:
7717:
7716:
7715:
7671:
7669:
7668:
7663:
7639:
7637:
7636:
7631:
7620:
7616:
7609:
7607:
7575:
7555:
7553:
7552:
7540:
7538:
7524:
7522:
7521:
7509:
7508:
7493:
7475:
7473:
7422:
7421:
7420:
7389:
7387:
7386:
7376:
7371:
7355:
7311:
7309:
7308:
7303:
7291:
7290:
7254:
7253:
7243:
7238:
7229:
7227:
7226:
7214:
7211:
7168:
7166:
7165:
7160:
7158:
7157:
7133:
7132:
7123:
7120:
7118:
7117:
7099:
7098:
7077:
7076:
7034:
7032:
7031:
7026:
7012:
7011:
6993:
6992:
6980:
6978:
6974:
6973:
6963:
6953:
6952:
6942:
6939:
6930:
6922:
6919:
6876:
6874:
6873:
6868:
6857:
6856:
6851:
6847:
6845:
6837:
6823:
6816:
6815:
6805:
6795:
6713:
6711:
6710:
6705:
6694:
6692:
6669:
6668:
6659:
6656:
6646:
6602:
6600:
6599:
6594:
6583:
6582:
6573:
6571:
6551:
6528:
6525:
6515:
6514:
6513:
6469:Le Roy summation
6453:
6451:
6450:
6445:
6434:
6432:
6431:
6430:
6408:
6407:
6406:
6384:
6382:
6381:
6371:
6355:
6354:
6353:
6318:Lambert summable
6282:
6280:
6279:
6274:
6263:
6259:
6251:
6245:
6237:
6235:
6234:
6224:
6202:
6158:
6156:
6155:
6152:
6149:
6130:
6128:
6127:
6124:
6121:
6080:Hölder summation
6074:Hölder summation
6058:
6056:
6055:
6050:
6048:
6043:
6042:
6033:
6024:
6022:
6021:
6016:
6014:
6006:
6001:
5996:
5995:
5986:
5977:
5975:
5974:
5969:
5932:
5930:
5929:
5924:
5909:
5904:
5879:
5877:
5876:
5871:
5859:
5857:
5856:
5851:
5839:
5837:
5836:
5831:
5795:
5793:
5792:
5787:
5778:
5776:
5753:
5752:
5751:
5739:
5738:
5728:
5723:
5722:
5709:
5704:
5667:
5665:
5664:
5659:
5647:
5645:
5637:
5636:
5635:
5626:
5625:
5615:
5610:
5609:
5596:
5591:
5536:For example, if
5511:
5509:
5508:
5503:
5454:
5452:
5451:
5446:
5438:
5436:
5435:
5426:
5425:
5424:
5415:
5414:
5404:
5399:
5397:
5396:
5387:
5386:
5385:
5376:
5375:
5365:
5321:
5319:
5318:
5313:
5304:
5303:
5288:
5287:
5213:
5211:
5210:
5205:
5200:
5199:
5181:
5180:
5168:
5167:
5154:
5153:
5144:
5136:
5125:
5110:
5107:
5099:
5085:
5084:
5081:
5046:Valiron's method
5002:and diverges at
4987:
4985:
4984:
4979:
4974:
4972:
4971:
4970:
4961:
4960:
4950:
4940:
4939:
4938:
4926:
4925:
4907:
4906:
4894:
4893:
4883:
4873:
4870:
4805:Bernoulli number
4792:
4790:
4789:
4784:
4778:
4776:
4765:
4764:
4749:
4735:
4734:
4722:
4721:
4709:
4708:
4696:
4695:
4685:
4680:
4637:
4630:
4629:
4627:
4626:
4623:
4620:
4604:
4598:
4596:
4595:
4592:
4589:
4514:zeta regularized
4483:
4481:
4480:
4475:
4467:
4464:
4459:
4447:
4442:
4439:
4434:
4422:
4417:
4414:
4409:
4397:
4342:
4340:
4339:
4334:
4326:
4324:
4323:
4314:
4313:
4304:
4299:
4297:
4296:
4287:
4286:
4277:
4272:
4270:
4269:
4260:
4259:
4250:
4216: = 1.
4191:
4189:
4188:
4182:
4179:
4155:
4153:
4152:
4147:
4142:
4137:
4114:
4100:
4099:
4090:
4089:
4084:
4078:
4076:
4068:
4067:
4066:
4054:
4039:
4037:
4036:
4017:
3986:
3984:
3983:
3978:
3962:
3960:
3959:
3954:
3942:
3940:
3939:
3934:
3860:
3859:
3858:
3849:
3840:
3833:
3831:
3814:
3812:
3811:
3786:
3761:
3759:
3758:
3753:
3741:
3739:
3738:
3733:
3731:
3723:
3658:
3656:
3655:
3650:
3618:
3616:
3615:
3610:
3605:
3597:
3556:
3554:
3553:
3548:
3528:
3526:
3525:
3520:
3512:
3511:
3496:
3495:
3477:
3476:
3464:
3463:
3445:
3443:
3442:
3441:
3410:
3409:
3408:
3374:
3369:
3367:
3366:
3365:
3349:
3348:
3347:
3331:
3320:
3318:
3317:
3312:
3232:
3201:Maclaurin series
3156:
3154:
3153:
3148:
3137:
3136:
3121:
3120:
3108:
3107:
3092:
3091:
3079:
3078:
3044:
3016:
2991:Cesàro summation
2985:
2983:
2982:
2977:
2972:
2971:
2962:
2961:
2951:
2946:
2930:
2929:
2928:
2884:) is defined as
2832:
2830:
2829:
2824:
2819:
2818:
2809:
2808:
2798:
2793:
2775:
2774:
2759:
2758:
2748:
2743:
2699:
2656:Dirichlet series
2635:
2633:
2632:
2627:
2609:
2608:
2607:
2575:
2574:
2536:
2534:
2533:
2528:
2526:
2525:
2521:
2520:
2503:
2502:
2492:
2487:
2447:
2437:
2383:
2371:Cesàro summation
2354:
2347:
2300:
2298:
2297:
2292:
2290:
2289:
2288:
2282:
2271:
2253:
2242:
2237:
2211:Cesàro summation
2170:
2168:
2167:
2162:
2160:
2158:
2157:
2156:
2138:
2137:
2125:
2124:
2114:
2113:
2112:
2103:
2102:
2084:
2083:
2074:
2073:
2055:
2054:
2045:
2044:
2034:
2029:
2028:
2002:
2000:
1999:
1994:
1986:
1984:
1983:
1982:
1964:
1963:
1951:
1950:
1940:
1939:
1930:
1896:
1878:
1853:
1800:
1798:
1797:
1792:
1790:
1787:
1786:
1784:
1781:
1776:
1774:
1760:
1729:
1726:
1723:
1684:
1680:
1677:
1674:
1672:
1671:
1658:
1653:
1623:
1619:
1616:
1613:
1611:
1610:
1591:
1586:
1559:
1556:
1555:
1553:
1552:
1539:
1534:
1480:geometric series
1477:
1432:
1361:
1339: >
1334:
1313:
1311:
1310:
1305:
1303:
1281:
1255:
1253:
1252:
1247:
1245:
1237:
1186:
1156:
1134:
1092:
1026:
983:
933:
918:
871:summation method
827:Fourier analysis
811:Cesàro summation
759:sums the series
753:partial converse
704:
702:
701:
696:
694:
686:
681:
678:
665:
663:
662:
657:
655:
647:
642:
639:
626:
624:
623:
618:
610:
607:
594:
592:
591:
586:
584:
576:
574:
571:
558:
556:
555:
550:
528:
526:
515:
514:
502:
499:
494:
483:
480:
467:
465:
464:
459:
457:
449:
447:
444:
431:
429:
428:
423:
421:
413:
411:
408:
382:Cesàro summation
359:
312:
310:
309:
306:
303:
293:
291:
290:
285:
242:Cesàro summation
238:partial function
234:summation method
213:
211:
210:
205:
200:
192:
189:
184:
160:
152:
147:
139:
134:
126:
121:
113:
63:divergent series
53:
21:
8702:
8701:
8697:
8696:
8695:
8693:
8692:
8691:
8657:
8656:
8655:
8650:
8632:
8589:
8538:Kinds of series
8529:
8468:
8435:Explicit series
8426:
8400:
8362:
8348:Cauchy sequence
8336:
8323:
8277:Figurate number
8254:
8248:
8239:Powers of three
8183:
8174:
8118:
8099:
8080:
8070:
8055:
8050:, North-Holland
8042:Brezinski, C.;
8041:
8031:
8021:
8018:
8013:
8012:
8004:
7997:
7989:
7982:
7974:
7967:
7959:
7955:
7915:
7914:
7910:
7900:
7899:
7895:
7887:
7883:
7875:
7871:
7845:
7844:
7840:
7829:"Translativity"
7827:
7826:
7822:
7809:
7808:
7804:
7799:
7787:
7750:
7739:
7729:
7707:
7691:
7690:
7684:
7678:
7645:
7644:
7576:
7556:
7544:
7528:
7513:
7500:
7499:
7495:
7423:
7412:
7390:
7378:
7336:
7335:
7326:
7318:
7276:
7245:
7218:
7192:
7191:
7186:
7143:
7124:
7121: for
7109:
7090:
7068:
7063:
7062:
7057:
7047:
7041:
7003:
6984:
6965:
6964:
6944:
6943:
6900:
6899:
6890:
6886:
6838:
6824:
6818:
6817:
6807:
6776:
6775:
6762:
6754:
6726:
6720:
6670:
6660:
6627:
6626:
6617:
6609:
6574:
6552:
6529:
6505:
6489:
6488:
6479:
6471:
6416:
6409:
6392:
6385:
6373:
6345:
6329:
6328:
6315:
6307:
6301:) summability.
6289:showed that if
6246:
6226:
6183:
6182:
6173:
6165:
6153:
6150:
6148:
6141:
6135:
6134:
6132:
6125:
6122:
6120:
6113:
6107:
6106:
6104:
6102:
6095:
6087:
6085:Hutton's method
6082:
6076:
6070:, chapter 11).
6065:
6034:
6027:
6026:
5987:
5980:
5979:
5942:
5941:
5885:
5884:
5862:
5861:
5842:
5841:
5822:
5821:
5814:
5809:
5754:
5740:
5730:
5729:
5711:
5691:
5690:
5685:
5638:
5627:
5617:
5616:
5598:
5578:
5577:
5571:Borel summation
5564:
5534:
5532:Borel summation
5523:
5475:
5474:
5427:
5416:
5406:
5405:
5388:
5377:
5367:
5366:
5344:
5343:
5338:
5331:
5325:are finite. If
5295:
5279:
5274:
5273:
5263:
5252:
5243:
5191:
5172:
5145:
5127:
5100:
5086:
5059:
5058:
5048:
5040:Borel summation
5024:
5022:Borel summation
4962:
4952:
4941:
4930:
4917:
4898:
4885:
4874:
4851:
4850:
4845:
4831:
4823:) = Σ
4813:
4801:
4766:
4750:
4726:
4713:
4700:
4687:
4643:
4642:
4638:and in general
4632:
4624:
4621:
4618:
4617:
4615:
4610:
4600:
4593:
4590:
4587:
4586:
4584:
4538:
4529:
4522:
4495:
4376:
4375:
4369:
4363:
4315:
4305:
4288:
4278:
4261:
4251:
4229:
4228:
4222:
4208:
4183:
4180:
4177:
4176:
4174:
4167:
4165:Euler summation
4161:
4159:Euler summation
4091:
4082:
4069:
4058:
4040:
4028:
3992:
3991:
3969:
3968:
3963:-functions and
3945:
3944:
3842:
3835:
3818:
3797:
3767:
3766:
3744:
3743:
3663:
3662:
3623:
3622:
3561:
3560:
3533:
3532:
3500:
3481:
3468:
3455:
3427:
3411:
3394:
3375:
3357:
3350:
3339:
3332:
3325:
3324:
3291:
3290:
3269:
3259:
3253:of a function.
3247:
3211:
3122:
3112:
3093:
3083:
3070:
3050:
3049:
3034:
3026:
3023:
3010:
2994:
2963:
2953:
2920:
2889:
2888:
2810:
2800:
2760:
2750:
2709:
2708:
2694:
2686:
2683:
2677:
2599:
2566:
2561:
2560:
2555:
2512:
2504:
2494:
2453:
2452:
2445:
2439:
2435:
2428:
2421:
2411:
2408:
2401:
2392:
2374:
2368:
2361:
2349:
2341:
2340:
2334:
2322:
2314:
2312:
2272:
2255:
2248:
2224:
2223:
2213:
2199:
2179:
2148:
2129:
2116:
2115:
2104:
2094:
2075:
2059:
2046:
2036:
2035:
2020:
2015:
2014:
1974:
1955:
1942:
1941:
1931:
1924:
1923:
1918:
1910:
1903:
1895:
1886:
1880:
1876:
1869:
1863:
1860:
1858:Sum of a series
1852:
1851:
1838:
1837:
1827:
1824:
1815:
1788:
1785:
1764:
1752:
1731:
1730:
1722:
1682:
1681:
1673:
1663:
1621:
1620:
1612:
1596:
1557:
1554:
1544:
1513:
1486:
1485:
1472:
1462:renormalization
1415:
1344:
1333:
1323:
1315:
1288:
1287:
1280:
1265:
1257:
1222:
1221:
1197:Borel summation
1176:
1162:
1155:
1145:
1136:
1124:
1110:
1091:
1084:
1074:
1065:
1059:
1025:
1016:
1006:
997:
948:
920:
905:
866:
850:renormalization
815:Borel summation
792:axiom of choice
730:Abelian theorem
711:
672:
671:
633:
632:
601:
600:
565:
564:
516:
503:
474:
473:
438:
437:
402:
401:
395:
361:
357:
341:
321:arithmetic mean
307:
304:
301:
300:
298:
252:
251:
101:
100:
95:harmonic series
67:infinite series
55:
48:
39:
28:
23:
22:
15:
12:
11:
5:
8700:
8698:
8690:
8689:
8684:
8679:
8674:
8669:
8659:
8658:
8652:
8651:
8649:
8648:
8637:
8634:
8633:
8631:
8630:
8625:
8620:
8615:
8610:
8605:
8599:
8597:
8591:
8590:
8588:
8587:
8582:
8580:Fourier series
8577:
8572:
8567:
8565:Puiseux series
8562:
8560:Laurent series
8557:
8552:
8547:
8541:
8539:
8535:
8534:
8531:
8530:
8528:
8527:
8522:
8517:
8512:
8507:
8502:
8497:
8492:
8487:
8482:
8476:
8474:
8470:
8469:
8467:
8466:
8461:
8456:
8451:
8445:
8443:
8436:
8432:
8431:
8428:
8427:
8425:
8424:
8419:
8414:
8408:
8406:
8402:
8401:
8399:
8398:
8393:
8388:
8383:
8377:
8375:
8368:
8364:
8363:
8361:
8360:
8355:
8350:
8344:
8342:
8338:
8337:
8330:
8328:
8325:
8324:
8322:
8321:
8320:
8319:
8309:
8304:
8299:
8294:
8289:
8284:
8279:
8274:
8269:
8263:
8261:
8250:
8249:
8247:
8246:
8241:
8236:
8231:
8226:
8221:
8216:
8211:
8206:
8200:
8198:
8191:
8185:
8184:
8175:
8173:
8172:
8165:
8158:
8150:
8144:
8143:
8140:
8137:
8134:
8116:
8097:
8078:
8068:
8053:
8039:
8029:
8017:
8014:
8011:
8010:
7995:
7980:
7978:, Appendix II.
7965:
7953:
7908:
7893:
7881:
7869:
7838:
7820:
7801:
7800:
7798:
7795:
7794:
7793:
7786:
7783:
7782:
7781:
7770:
7767:
7764:
7757:
7753:
7749:
7746:
7742:
7736:
7732:
7726:
7722:
7714:
7710:
7706:
7703:
7699:
7680:Main article:
7677:
7674:
7661:
7658:
7655:
7652:
7641:
7640:
7629:
7626:
7623:
7619:
7615:
7612:
7606:
7603:
7600:
7597:
7594:
7591:
7588:
7585:
7582:
7579:
7574:
7571:
7568:
7565:
7562:
7559:
7551:
7547:
7543:
7537:
7534:
7531:
7527:
7520:
7516:
7512:
7507:
7503:
7498:
7492:
7489:
7486:
7482:
7478:
7472:
7469:
7466:
7463:
7460:
7457:
7454:
7451:
7447:
7444:
7441:
7438:
7435:
7432:
7429:
7426:
7419:
7415:
7411:
7408:
7405:
7402:
7399:
7396:
7393:
7385:
7381:
7375:
7370:
7367:
7364:
7360:
7354:
7351:
7348:
7344:
7324:
7317:
7314:
7313:
7312:
7301:
7298:
7295:
7289:
7286:
7283:
7279:
7275:
7272:
7269:
7266:
7263:
7260:
7257:
7252:
7248:
7242:
7237:
7233:
7225:
7221:
7217:
7210:
7207:
7204:
7200:
7184:
7170:
7169:
7156:
7153:
7150:
7146:
7142:
7139:
7136:
7131:
7127:
7116:
7112:
7108:
7105:
7102:
7097:
7093:
7089:
7086:
7083:
7080:
7075:
7071:
7053:
7043:Main article:
7040:
7037:
7036:
7035:
7024:
7021:
7018:
7015:
7010:
7006:
7002:
6999:
6996:
6991:
6987:
6983:
6977:
6972:
6968:
6962:
6959:
6956:
6951:
6947:
6938:
6934:
6928:
6925:
6918:
6915:
6912:
6908:
6888:
6884:
6878:
6877:
6866:
6863:
6860:
6855:
6850:
6844:
6841:
6836:
6833:
6830:
6827:
6821:
6814:
6810:
6804:
6800:
6794:
6791:
6788:
6784:
6760:
6753:
6750:
6722:Main article:
6719:
6716:
6715:
6714:
6703:
6700:
6697:
6691:
6688:
6685:
6682:
6679:
6676:
6673:
6667:
6663:
6655:
6651:
6645:
6642:
6639:
6635:
6615:
6608:
6605:
6604:
6603:
6592:
6589:
6586:
6581:
6577:
6570:
6567:
6564:
6561:
6558:
6555:
6550:
6547:
6544:
6541:
6538:
6535:
6532:
6524:
6520:
6512:
6508:
6504:
6501:
6497:
6477:
6470:
6467:
6455:
6454:
6443:
6440:
6437:
6429:
6426:
6423:
6419:
6415:
6412:
6405:
6402:
6399:
6395:
6391:
6388:
6380:
6376:
6370:
6367:
6364:
6360:
6352:
6348:
6344:
6341:
6337:
6313:
6306:
6303:
6284:
6283:
6272:
6269:
6266:
6262:
6257:
6254:
6249:
6243:
6240:
6233:
6229:
6223:
6220:
6217:
6214:
6211:
6207:
6201:
6198:
6195:
6191:
6171:
6164:
6161:
6146:
6139:
6118:
6111:
6100:
6093:
6086:
6083:
6078:Main article:
6075:
6072:
6064:
6061:
6046:
6041:
6037:
6012:
6009:
6004:
5999:
5994:
5990:
5967:
5964:
5961:
5958:
5955:
5952:
5949:
5934:
5933:
5922:
5919:
5916:
5913:
5908:
5903:
5900:
5897:
5893:
5869:
5849:
5829:
5813:
5810:
5808:
5805:
5797:
5796:
5785:
5782:
5775:
5772:
5769:
5766:
5763:
5760:
5757:
5750:
5747:
5743:
5737:
5733:
5726:
5721:
5718:
5714:
5708:
5703:
5699:
5683:
5669:
5668:
5657:
5654:
5651:
5644:
5641:
5634:
5630:
5624:
5620:
5613:
5608:
5605:
5601:
5595:
5590:
5586:
5560:
5533:
5530:
5519:
5513:
5512:
5501:
5498:
5494:
5491:
5488:
5485:
5482:
5456:
5455:
5444:
5441:
5434:
5430:
5423:
5419:
5413:
5409:
5402:
5395:
5391:
5384:
5380:
5374:
5370:
5363:
5360:
5357:
5354:
5351:
5336:
5329:
5323:
5322:
5311:
5308:
5302:
5298:
5294:
5291:
5286:
5282:
5262:
5261:Moment methods
5259:
5248:
5241:
5233:) =
5215:
5214:
5203:
5198:
5194:
5190:
5187:
5184:
5179:
5175:
5171:
5166:
5163:
5160:
5157:
5152:
5148:
5142:
5139:
5134:
5130:
5124:
5121:
5118:
5114:
5106:
5103:
5098:
5095:
5092:
5089:
5080:
5077:
5074:
5071:
5067:
5047:
5044:
5034:) =
5023:
5020:
4989:
4988:
4977:
4969:
4965:
4959:
4955:
4949:
4945:
4937:
4933:
4929:
4924:
4920:
4916:
4913:
4910:
4905:
4901:
4897:
4892:
4888:
4882:
4878:
4869:
4866:
4863:
4859:
4843:
4827:
4812:
4809:
4799:
4794:
4793:
4782:
4775:
4772:
4769:
4763:
4760:
4757:
4753:
4747:
4744:
4741:
4738:
4733:
4729:
4725:
4720:
4716:
4712:
4707:
4703:
4699:
4694:
4690:
4684:
4679:
4676:
4673:
4669:
4665:
4662:
4659:
4656:
4653:
4650:
4534:
4527:
4520:
4491:
4485:
4484:
4473:
4470:
4463:
4458:
4454:
4450:
4445:
4438:
4433:
4429:
4425:
4420:
4413:
4408:
4404:
4400:
4395:
4392:
4389:
4386:
4383:
4371:If the series
4365:Main article:
4362:
4359:
4344:
4343:
4332:
4329:
4322:
4318:
4312:
4308:
4302:
4295:
4291:
4285:
4281:
4275:
4268:
4264:
4258:
4254:
4248:
4245:
4242:
4239:
4236:
4221:
4218:
4204:
4187: + 1
4163:Main article:
4160:
4157:
4145:
4140:
4135:
4132:
4129:
4126:
4123:
4120:
4117:
4113:
4109:
4106:
4103:
4098:
4094:
4088:
4081:
4075:
4072:
4065:
4061:
4057:
4053:
4049:
4046:
4043:
4035:
4031:
4027:
4024:
4021:
4016:
4013:
4010:
4006:
4002:
3999:
3976:
3952:
3932:
3929:
3926:
3923:
3920:
3917:
3914:
3911:
3908:
3905:
3902:
3899:
3896:
3893:
3890:
3887:
3884:
3881:
3878:
3875:
3872:
3869:
3866:
3863:
3857:
3852:
3848:
3845:
3839:
3830:
3827:
3824:
3821:
3817:
3810:
3807:
3804:
3800:
3796:
3793:
3790:
3785:
3782:
3779:
3775:
3751:
3729:
3726:
3721:
3718:
3715:
3712:
3709:
3706:
3703:
3700:
3697:
3694:
3691:
3688:
3685:
3682:
3679:
3676:
3673:
3670:
3648:
3645:
3642:
3639:
3636:
3633:
3630:
3608:
3603:
3600:
3595:
3592:
3589:
3586:
3583:
3580:
3577:
3574:
3571:
3568:
3546:
3543:
3540:
3531:Evaluating at
3518:
3515:
3510:
3507:
3503:
3499:
3494:
3491:
3488:
3484:
3480:
3475:
3471:
3467:
3462:
3458:
3454:
3451:
3448:
3440:
3437:
3434:
3430:
3426:
3423:
3420:
3417:
3414:
3407:
3404:
3401:
3397:
3393:
3390:
3387:
3384:
3381:
3378:
3372:
3364:
3360:
3356:
3353:
3346:
3342:
3338:
3335:
3310:
3307:
3304:
3301:
3298:
3265:
3258:
3255:
3246:
3243:
3158:
3157:
3146:
3143:
3140:
3135:
3132:
3129:
3125:
3119:
3115:
3111:
3106:
3103:
3100:
3096:
3090:
3086:
3082:
3077:
3073:
3069:
3066:
3063:
3060:
3057:
3030:
3022:
3019:
3006:
2987:
2986:
2975:
2970:
2966:
2960:
2956:
2950:
2945:
2942:
2939:
2935:
2927:
2923:
2919:
2916:
2912:
2908:
2905:
2902:
2899:
2896:
2858:positive reals
2834:
2833:
2822:
2817:
2813:
2807:
2803:
2797:
2792:
2789:
2786:
2782:
2778:
2773:
2770:
2767:
2763:
2757:
2753:
2747:
2742:
2739:
2736:
2732:
2728:
2725:
2722:
2719:
2716:
2702:Abel summation
2690:
2681:Abel's theorem
2676:
2675:Abel summation
2673:
2637:
2636:
2625:
2622:
2619:
2616:
2613:
2606:
2602:
2598:
2595:
2591:
2587:
2584:
2581:
2578:
2573:
2569:
2556:is defined as
2551:
2538:
2537:
2524:
2519:
2515:
2511:
2507:
2501:
2497:
2491:
2486:
2483:
2480:
2476:
2472:
2469:
2466:
2463:
2460:
2443:
2433:
2426:
2419:
2407:
2404:
2397:
2388:
2366:
2359:
2338:
2332:
2331:
2318:
2313:is defined by
2308:
2302:
2301:
2287:
2281:
2278:
2275:
2270:
2267:
2264:
2261:
2258:
2252:
2246:
2241:
2236:
2232:
2212:
2209:
2195:
2177:
2172:
2171:
2155:
2151:
2147:
2144:
2141:
2136:
2132:
2128:
2123:
2119:
2111:
2107:
2101:
2097:
2093:
2090:
2087:
2082:
2078:
2072:
2069:
2066:
2062:
2058:
2053:
2049:
2043:
2039:
2032:
2027:
2023:
2004:
2003:
1992:
1989:
1981:
1977:
1973:
1970:
1967:
1962:
1958:
1954:
1949:
1945:
1938:
1934:
1916:
1908:
1902:
1899:
1891:
1884:
1874:
1867:
1859:
1856:
1847:
1843:
1835:
1831:
1823:
1820:
1814:
1811:
1802:
1801:
1779:
1773:
1770:
1767:
1763:
1758:
1755:
1753:
1751:
1748:
1745:
1742:
1739:
1736:
1733:
1732:
1724:
1721:
1718:
1715:
1712:
1709:
1706:
1703:
1699:
1696:
1693:
1690:
1687:
1685:
1683:
1675:
1670:
1666:
1662:
1657:
1652:
1649:
1646:
1642:
1638:
1635:
1632:
1629:
1626:
1624:
1622:
1614:
1609:
1606:
1603:
1599:
1595:
1590:
1585:
1582:
1579:
1575:
1571:
1568:
1565:
1562:
1560:
1558:
1551:
1547:
1543:
1538:
1533:
1530:
1527:
1523:
1519:
1516:
1514:
1512:
1509:
1506:
1503:
1500:
1497:
1494:
1493:
1384:
1383:
1378:also exhibits
1329:
1319:
1302:
1298:
1295:
1271:
1261:
1244:
1240:
1236:
1232:
1229:
1193:
1192:
1174:
1150:
1141:
1122:
1089:
1079:
1070:
1057:
1040:
1027:of the series
1021:
1011:
1001:
984:for sequences
934:
865:
862:
823:Banach algebra
807:Abel summation
755:means that if
739:Abel's theorem
710:
707:
706:
705:
692:
689:
684:
666:
653:
650:
645:
627:
616:
613:
595:
582:
579:
559:
548:
545:
541:
538:
535:
532:
525:
522:
519:
513:
510:
506:
498:
493:
489:
468:
455:
452:
432:
419:
416:
394:
391:
378:Ernesto Cesàro
374:Henri Poincaré
365:Leonhard Euler
342:
340:
337:
333:regularization
295:
294:
283:
280:
277:
274:
271:
268:
265:
262:
259:
215:
214:
203:
198:
195:
188:
183:
180:
177:
173:
169:
166:
163:
158:
155:
150:
145:
142:
137:
132:
129:
124:
119:
116:
111:
108:
91:counterexample
40:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8699:
8688:
8685:
8683:
8680:
8678:
8675:
8673:
8670:
8668:
8665:
8664:
8662:
8647:
8639:
8638:
8635:
8629:
8626:
8624:
8621:
8619:
8616:
8614:
8611:
8609:
8606:
8604:
8601:
8600:
8598:
8596:
8592:
8586:
8583:
8581:
8578:
8576:
8573:
8571:
8568:
8566:
8563:
8561:
8558:
8556:
8553:
8551:
8548:
8546:
8545:Taylor series
8543:
8542:
8540:
8536:
8526:
8523:
8521:
8518:
8516:
8513:
8511:
8508:
8506:
8503:
8501:
8498:
8496:
8493:
8491:
8488:
8486:
8483:
8481:
8478:
8477:
8475:
8471:
8465:
8462:
8460:
8457:
8455:
8452:
8450:
8447:
8446:
8444:
8440:
8437:
8433:
8423:
8420:
8418:
8415:
8413:
8410:
8409:
8407:
8403:
8397:
8394:
8392:
8389:
8387:
8384:
8382:
8379:
8378:
8376:
8372:
8369:
8365:
8359:
8356:
8354:
8351:
8349:
8346:
8345:
8343:
8339:
8334:
8318:
8315:
8314:
8313:
8310:
8308:
8305:
8303:
8300:
8298:
8295:
8293:
8290:
8288:
8285:
8283:
8280:
8278:
8275:
8273:
8270:
8268:
8265:
8264:
8262:
8258:
8251:
8245:
8242:
8240:
8237:
8235:
8234:Powers of two
8232:
8230:
8227:
8225:
8222:
8220:
8219:Square number
8217:
8215:
8212:
8210:
8207:
8205:
8202:
8201:
8199:
8195:
8192:
8190:
8186:
8182:
8178:
8171:
8166:
8164:
8159:
8157:
8152:
8151:
8148:
8141:
8138:
8135:
8131:
8127:
8126:
8121:
8117:
8113:
8109:
8108:
8103:
8098:
8094:
8090:
8089:
8084:
8079:
8074:
8069:
8064:
8063:
8058:
8054:
8049:
8045:
8040:
8035:
8030:
8025:
8020:
8019:
8015:
8007:
8002:
8000:
7996:
7992:
7987:
7985:
7981:
7977:
7972:
7970:
7966:
7962:
7957:
7954:
7949:
7945:
7941:
7937:
7932:
7927:
7923:
7919:
7912:
7909:
7904:
7897:
7894:
7891:, p. 14.
7890:
7885:
7882:
7878:
7873:
7870:
7865:
7861:
7857:
7853:
7849:
7842:
7839:
7834:
7830:
7824:
7821:
7816:
7812:
7806:
7803:
7796:
7792:
7789:
7788:
7784:
7768:
7765:
7762:
7755:
7751:
7747:
7744:
7740:
7734:
7730:
7724:
7720:
7712:
7708:
7701:
7689:
7688:
7687:
7683:
7675:
7673:
7656:
7627:
7624:
7621:
7617:
7613:
7610:
7601:
7598:
7595:
7586:
7583:
7580:
7569:
7566:
7563:
7557:
7549:
7545:
7541:
7535:
7532:
7529:
7525:
7518:
7514:
7510:
7505:
7501:
7496:
7484:
7476:
7467:
7464:
7461:
7458:
7455:
7442:
7439:
7436:
7433:
7430:
7417:
7406:
7403:
7400:
7383:
7379:
7373:
7368:
7365:
7362:
7358:
7346:
7334:
7333:
7332:
7330:
7323:
7315:
7299:
7296:
7293:
7287:
7284:
7281:
7273:
7270:
7267:
7258:
7250:
7246:
7240:
7235:
7231:
7223:
7219:
7215:
7202:
7190:
7189:
7188:
7183:
7179:
7175:
7154:
7151:
7148:
7144:
7140:
7137:
7134:
7129:
7125:
7114:
7110:
7106:
7103:
7100:
7095:
7091:
7087:
7081:
7073:
7069:
7061:
7060:
7059:
7056:
7052:
7046:
7038:
7022:
7019:
7016:
7008:
7004:
7000:
6997:
6994:
6989:
6985:
6975:
6970:
6966:
6960:
6957:
6954:
6949:
6945:
6936:
6932:
6926:
6923:
6916:
6910:
6898:
6897:
6896:
6894:
6883:
6864:
6861:
6858:
6853:
6848:
6842:
6839:
6834:
6831:
6828:
6825:
6819:
6812:
6808:
6802:
6798:
6792:
6786:
6774:
6773:
6772:
6770:
6766:
6759:
6751:
6749:
6747:
6743:
6739:
6735:
6731:
6725:
6717:
6701:
6698:
6695:
6686:
6683:
6680:
6677:
6665:
6661:
6653:
6649:
6643:
6637:
6625:
6624:
6623:
6621:
6614:
6606:
6590:
6587:
6584:
6579:
6575:
6565:
6562:
6559:
6545:
6542:
6539:
6536:
6522:
6518:
6510:
6506:
6499:
6487:
6486:
6485:
6483:
6476:
6468:
6466:
6464:
6460:
6441:
6438:
6435:
6427:
6424:
6421:
6417:
6413:
6410:
6403:
6400:
6397:
6393:
6389:
6386:
6378:
6374:
6368:
6365:
6362:
6358:
6350:
6346:
6339:
6327:
6326:
6325:
6323:
6319:
6312:
6304:
6302:
6300:
6296:
6292:
6288:
6287:Albert Ingham
6270:
6267:
6264:
6260:
6255:
6252:
6247:
6241:
6238:
6231:
6227:
6221:
6218:
6215:
6212:
6209:
6205:
6193:
6181:
6180:
6179:
6177:
6170:
6162:
6160:
6145:
6142: +
6138:
6117:
6114: +
6110:
6099:
6092:
6081:
6073:
6071:
6069:
6062:
6060:
6044:
6039:
6035:
6010:
6007:
6002:
5997:
5992:
5988:
5965:
5962:
5959:
5956:
5953:
5950:
5947:
5939:
5917:
5911:
5906:
5901:
5898:
5895:
5891:
5883:
5882:
5881:
5867:
5827:
5819:
5811:
5806:
5804:
5802:
5783:
5780:
5770:
5767:
5764:
5761:
5748:
5745:
5741:
5735:
5731:
5724:
5719:
5716:
5712:
5701:
5697:
5689:
5688:
5687:
5682:
5678:
5674:
5655:
5652:
5649:
5642:
5639:
5632:
5628:
5622:
5618:
5611:
5606:
5603:
5599:
5588:
5584:
5576:
5575:
5574:
5572:
5568:
5565: =
5563:
5559:
5555:
5551:
5548:for positive
5547:
5543:
5540: =
5539:
5531:
5529:
5527:
5522:
5518:
5499:
5496:
5489:
5483:
5480:
5473:
5472:
5471:
5469:
5465:
5461:
5442:
5439:
5432:
5428:
5421:
5417:
5411:
5407:
5400:
5393:
5389:
5382:
5378:
5372:
5368:
5361:
5355:
5349:
5342:
5341:
5340:
5335:
5332: +
5328:
5309:
5306:
5300:
5296:
5292:
5289:
5284:
5280:
5272:
5271:
5270:
5268:
5265:Suppose that
5260:
5258:
5256:
5251:
5247:
5240:
5236:
5232:
5228:
5224:
5220:
5196:
5192:
5188:
5185:
5182:
5177:
5173:
5161:
5155:
5150:
5146:
5140:
5137:
5132:
5128:
5122:
5119:
5116:
5112:
5104:
5101:
5093:
5087:
5075:
5069:
5057:
5056:
5055:
5053:
5043:
5041:
5037:
5033:
5029:
5021:
5019:
5017:
5013:
5009:
5006: =
5005:
5001:
4997:
4992:
4975:
4967:
4963:
4957:
4953:
4947:
4943:
4935:
4931:
4922:
4918:
4914:
4911:
4908:
4903:
4899:
4890:
4886:
4880:
4876:
4861:
4849:
4848:
4847:
4842:
4838:
4834:
4830:
4826:
4822:
4818:
4810:
4808:
4806:
4802:
4780:
4773:
4770:
4767:
4761:
4758:
4755:
4751:
4745:
4742:
4739:
4736:
4731:
4727:
4723:
4718:
4714:
4710:
4705:
4701:
4697:
4692:
4688:
4677:
4674:
4671:
4667:
4663:
4657:
4654:
4648:
4641:
4640:
4639:
4635:
4613:
4608:
4603:
4582:
4578:
4574:
4570:
4566:
4562:
4558:
4554:
4550:
4546:
4542:
4537:
4533:
4526:
4523: +
4519:
4515:
4511:
4507:
4503:
4499:
4494:
4490:
4471:
4468:
4461:
4456:
4452:
4448:
4443:
4436:
4431:
4427:
4423:
4418:
4411:
4406:
4402:
4398:
4393:
4387:
4381:
4374:
4373:
4372:
4368:
4360:
4358:
4356:
4351:
4349:
4330:
4327:
4320:
4316:
4310:
4306:
4300:
4293:
4289:
4283:
4279:
4273:
4266:
4262:
4256:
4252:
4246:
4240:
4234:
4227:
4226:
4225:
4219:
4217:
4215:
4210:
4207:
4203:
4199:
4195:
4186:
4172:
4166:
4158:
4156:
4143:
4138:
4133:
4127:
4124:
4121:
4118:
4115:
4111:
4107:
4104:
4096:
4092:
4086:
4079:
4073:
4070:
4063:
4055:
4051:
4047:
4044:
4033:
4025:
4022:
4014:
4011:
4008:
4004:
4000:
3997:
3990:
3966:
3930:
3927:
3924:
3921:
3918:
3915:
3912:
3909:
3906:
3903:
3900:
3897:
3894:
3891:
3888:
3885:
3882:
3879:
3876:
3873:
3870:
3867:
3864:
3861:
3850:
3846:
3843:
3828:
3825:
3822:
3819:
3815:
3808:
3805:
3802:
3794:
3791:
3783:
3780:
3777:
3773:
3763:
3749:
3727:
3724:
3719:
3716:
3713:
3710:
3707:
3704:
3701:
3698:
3695:
3692:
3689:
3686:
3683:
3680:
3677:
3674:
3671:
3668:
3660:
3646:
3643:
3640:
3637:
3634:
3631:
3628:
3619:
3606:
3601:
3598:
3593:
3590:
3587:
3584:
3581:
3578:
3575:
3572:
3569:
3566:
3558:
3544:
3541:
3538:
3529:
3516:
3513:
3508:
3505:
3501:
3497:
3492:
3489:
3486:
3482:
3478:
3473:
3469:
3465:
3460:
3456:
3452:
3449:
3446:
3438:
3435:
3432:
3428:
3424:
3421:
3418:
3415:
3412:
3405:
3402:
3399:
3395:
3391:
3388:
3385:
3382:
3379:
3376:
3370:
3362:
3358:
3354:
3351:
3344:
3340:
3336:
3333:
3322:
3308:
3305:
3302:
3299:
3296:
3288:
3284:
3280:
3276:
3272:
3268:
3264:
3256:
3254:
3252:
3244:
3242:
3240:
3236:
3230:
3226:
3222:
3218:
3214:
3209:
3205:
3202:
3198:
3194:
3189:
3187:
3183:
3179:
3175:
3171:
3167:
3163:
3144:
3141:
3138:
3133:
3130:
3127:
3123:
3117:
3113:
3109:
3104:
3101:
3098:
3094:
3088:
3084:
3080:
3075:
3071:
3067:
3061:
3055:
3048:
3047:
3046:
3042:
3038:
3033:
3029:
3020:
3018:
3014:
3009:
3005:
3001:
2997:
2992:
2973:
2968:
2964:
2958:
2954:
2943:
2940:
2937:
2933:
2925:
2921:
2914:
2906:
2900:
2894:
2887:
2886:
2885:
2883:
2879:
2875:
2871:
2867:
2863:
2859:
2855:
2851:
2847:
2843:
2839:
2820:
2815:
2811:
2805:
2801:
2790:
2787:
2784:
2780:
2776:
2771:
2768:
2765:
2761:
2755:
2751:
2740:
2737:
2734:
2730:
2726:
2720:
2714:
2707:
2706:
2705:
2703:
2698:
2693:
2689:
2682:
2674:
2672:
2670:
2665:
2663:
2662:
2657:
2652:
2650:
2646:
2642:
2623:
2617:
2611:
2604:
2600:
2593:
2585:
2579:
2571:
2567:
2559:
2558:
2557:
2554:
2550:
2547:
2543:
2522:
2517:
2513:
2509:
2505:
2499:
2495:
2484:
2481:
2478:
2474:
2470:
2464:
2458:
2451:
2450:
2449:
2442:
2432:
2425:
2418:
2414:
2406:Abelian means
2405:
2403:
2400:
2396:
2391:
2387:
2381:
2377:
2372:
2365:
2358:
2352:
2345:
2337:
2330:
2326:
2321:
2317:
2311:
2307:
2279:
2276:
2273:
2268:
2265:
2262:
2259:
2256:
2244:
2239:
2234:
2230:
2222:
2221:
2220:
2218:
2210:
2208:
2205:
2203:
2198:
2194:
2191:
2189:
2184:
2180:
2153:
2149:
2145:
2142:
2139:
2134:
2130:
2126:
2121:
2117:
2109:
2105:
2099:
2095:
2091:
2088:
2085:
2080:
2076:
2070:
2067:
2064:
2060:
2056:
2051:
2047:
2041:
2037:
2030:
2025:
2021:
2013:
2012:
2011:
2009:
1990:
1979:
1975:
1971:
1968:
1965:
1960:
1956:
1952:
1947:
1943:
1936:
1932:
1922:
1921:
1920:
1915:
1911:
1901:Nørlund means
1900:
1898:
1894:
1890:
1883:
1873:
1866:
1857:
1855:
1850:
1846:
1842:
1834:
1830:
1821:
1819:
1812:
1810:
1808:
1777:
1771:
1768:
1765:
1761:
1756:
1754:
1746:
1743:
1740:
1734:
1719:
1713:
1710:
1707:
1701:
1697:
1694:
1691:
1688:
1686:
1668:
1664:
1660:
1650:
1647:
1644:
1640:
1636:
1633:
1630:
1627:
1625:
1607:
1604:
1601:
1597:
1593:
1583:
1580:
1577:
1573:
1569:
1566:
1563:
1561:
1549:
1545:
1541:
1531:
1528:
1525:
1521:
1517:
1515:
1507:
1504:
1501:
1495:
1484:
1483:
1482:
1481:
1475:
1469:
1465:
1463:
1459:
1455:
1451:
1446:
1444:
1440:
1436:
1430:
1426:
1422:
1418:
1413:
1409:
1405:
1401:
1397:
1393:
1389:
1381:
1377:
1373:
1369:
1365:
1359:
1355:
1351:
1347:
1342:
1338:
1332:
1327:
1322:
1318:
1296:
1293:
1285:
1278:
1274:
1269:
1264:
1260:
1230:
1227:
1220:
1216:
1212:
1208:
1205:
1204:
1203:
1200:
1198:
1190:
1184:
1180:
1173:
1169:
1165:
1160:
1153:
1149:
1144:
1139:
1132:
1128:
1121:
1117:
1113:
1108:
1104:
1100:
1096:
1088:
1082:
1078:
1073:
1068:
1063:
1056:
1052:
1048:
1047:translativity
1045:(also called
1044:
1041:
1038:
1034:
1030:
1024:
1020:
1014:
1010:
1004:
1000:
995:
991:
987:
981:
977:
973:
969:
966:
962:
958:
955:
951:
946:
942:
938:
935:
931:
927:
923:
916:
912:
908:
903:
900:converges to
899:
895:
891:
888:
887:
886:
883:
880:
876:
872:
863:
861:
859:
855:
851:
847:
843:
839:
835:
834:extrapolation
830:
828:
824:
820:
816:
812:
808:
804:
799:
797:
793:
789:
788:
783:
779:
778:
772:
770:
766:
762:
758:
754:
750:
749:Alfred Tauber
746:
745:
740:
736:
732:
731:
726:
722:
721:
716:
708:
690:
687:
682:
670:
667:
651:
648:
643:
631:
628:
614:
611:
599:
596:
580:
577:
563:
560:
546:
543:
539:
536:
533:
530:
523:
520:
517:
511:
508:
504:
491:
487:
472:
469:
453:
450:
436:
433:
417:
414:
400:
397:
396:
392:
390:
387:
383:
379:
375:
371:
366:
360:
354:
352:
348:
338:
336:
334:
330:
326:
322:
318:
317:
281:
278:
275:
272:
269:
266:
263:
260:
257:
250:
249:
248:
247:
243:
239:
235:
231:
226:
224:
223:Nicole Oresme
220:
201:
196:
193:
181:
178:
175:
171:
167:
164:
161:
156:
153:
148:
143:
140:
135:
130:
127:
122:
117:
114:
109:
106:
99:
98:
97:
96:
92:
86:
84:
80:
76:
72:
68:
64:
60:
54:
51:
45:
37:
33:
19:
8550:Power series
8390:
8292:Lucas number
8244:Powers of 10
8224:Cubic number
8123:
8105:
8086:
8072:
8061:
8057:Hardy, G. H.
8047:
8033:
8023:
7956:
7921:
7917:
7911:
7896:
7884:
7872:
7855:
7851:
7847:
7841:
7832:
7823:
7814:
7805:
7685:
7642:
7328:
7321:
7319:
7181:
7177:
7173:
7171:
7054:
7050:
7048:
6892:
6891:summable to
6881:
6879:
6768:
6764:
6757:
6755:
6745:
6741:
6737:
6733:
6727:
6619:
6612:
6610:
6481:
6474:
6472:
6462:
6458:
6456:
6321:
6310:
6308:
6298:
6294:
6290:
6285:
6175:
6168:
6166:
6143:
6136:
6115:
6108:
6097:
6090:
6088:
6066:
5935:
5815:
5800:
5798:
5680:
5676:
5672:
5670:
5566:
5561:
5557:
5553:
5549:
5545:
5541:
5537:
5535:
5525:
5520:
5516:
5514:
5467:
5466:, then the (
5463:
5459:
5457:
5333:
5326:
5324:
5266:
5264:
5254:
5249:
5245:
5238:
5234:
5230:
5226:
5222:
5218:
5216:
5051:
5049:
5035:
5031:
5027:
5025:
5015:
5011:
5007:
5003:
4999:
4995:
4993:
4990:
4840:
4836:
4832:
4828:
4824:
4820:
4816:
4814:
4797:
4795:
4633:
4611:
4606:
4580:
4576:
4572:
4564:
4560:
4556:
4552:
4548:
4544:
4540:
4535:
4531:
4524:
4517:
4509:
4505:
4497:
4492:
4488:
4486:
4370:
4354:
4352:
4347:
4345:
4223:
4213:
4211:
4205:
4201:
4197:
4193:
4184:
4170:
4168:
3965:Pochhammer's
3764:
3661:
3620:
3559:
3530:
3323:
3286:
3282:
3278:
3274:
3270:
3266:
3262:
3260:
3248:
3238:
3234:
3228:
3224:
3220:
3216:
3212:
3207:
3203:
3196:
3192:
3190:
3181:
3177:
3173:
3170:Lindelöf sum
3169:
3165:
3161:
3159:
3040:
3036:
3031:
3027:
3024:
3012:
3007:
3003:
2999:
2995:
2988:
2881:
2877:
2873:
2869:
2865:
2862:power series
2853:
2849:
2845:
2841:
2837:
2835:
2701:
2696:
2691:
2687:
2684:
2668:
2666:
2659:
2653:
2648:
2644:
2640:
2638:
2552:
2548:
2546:Abelian mean
2545:
2541:
2539:
2440:
2430:
2423:
2416:
2412:
2409:
2398:
2394:
2389:
2385:
2379:
2375:
2369:is ordinary
2363:
2356:
2350:
2343:
2335:
2328:
2324:
2319:
2315:
2309:
2305:
2303:
2216:
2214:
2206:
2201:
2196:
2192:
2186:
2182:
2175:
2173:
2007:
2005:
1913:
1906:
1904:
1892:
1888:
1881:
1871:
1864:
1861:
1848:
1844:
1840:
1832:
1828:
1825:
1816:
1806:
1803:
1473:
1470:
1466:
1464:techniques.
1447:
1442:
1438:
1434:
1428:
1424:
1420:
1416:
1411:
1403:
1399:
1395:
1394:to share is
1391:
1387:
1385:
1379:
1375:
1371:
1367:
1363:
1357:
1353:
1349:
1345:
1340:
1336:
1330:
1325:
1320:
1316:
1283:
1276:
1272:
1267:
1262:
1258:
1214:
1210:
1206:
1201:
1194:
1182:
1178:
1171:
1167:
1163:
1158:
1151:
1147:
1142:
1137:
1130:
1126:
1119:
1115:
1111:
1106:
1102:
1098:
1094:
1086:
1080:
1076:
1071:
1066:
1061:
1054:
1050:
1046:
1042:
1036:
1032:
1028:
1022:
1018:
1012:
1008:
1002:
998:
993:
989:
985:
979:
975:
971:
967:
964:
960:
956:
953:
949:
944:
940:
936:
929:
925:
921:
914:
910:
906:
901:
897:
893:
889:
881:
878:
874:
870:
867:
836:methods and
831:
800:
796:Zorn's lemma
787:Banach limit
785:
775:
773:
768:
764:
760:
756:
752:
742:
734:
728:
718:
714:
712:
362:
356:
350:
346:
344:
314:
296:
233:
229:
227:
216:
87:
79:partial sums
69:that is not
62:
56:
47:
42:
8417:Conditional
8405:Convergence
8396:Telescoping
8381:Alternating
8297:Pell number
7924:(1): 7–15.
7877:Volkov 2001
7835:. Springer.
7320:The series
7039:Riesz means
6880:The series
6756:The series
6611:The series
6473:The series
6309:The series
6167:The series
6068:Hardy (1949
4500:and can be
3557:, one gets
1396:consistency
825:methods in
59:mathematics
18:Summability
8661:Categories
8442:Convergent
8386:Convergent
8016:References
8006:Hardy 1949
7991:Hardy 1949
7976:Hardy 1949
7961:Hardy 1949
7931:1804.11342
7889:Hardy 1949
7045:Riesz mean
2679:See also:
2448:. Suppose
1408:consistent
1314:such that
1256:such that
1189:shift rule
890:Regularity
297:the value
219:was proven
71:convergent
50:N. H. Abel
8473:Divergent
8391:Divergent
8253:Advanced
8229:Factorial
8177:Sequences
8130:EMS Press
8112:EMS Press
8093:EMS Press
7948:119665957
7748:α
7745:−
7721:∑
7705:→
7702:α
7651:Γ
7614:⋯
7567:−
7491:∞
7488:→
7450:Γ
7440:−
7425:Γ
7395:Γ
7359:∑
7353:∞
7350:→
7285:−
7282:κ
7271:−
7268:ω
7251:λ
7241:ω
7232:∫
7224:κ
7220:ω
7216:κ
7209:∞
7206:→
7203:ω
7145:λ
7141:≤
7126:λ
7104:⋯
7074:λ
6998:⋯
6955:
6933:∑
6927:π
6914:→
6829:
6799:∑
6790:→
6684:δ
6672:Γ
6650:∑
6641:→
6638:δ
6554:Γ
6543:ζ
6531:Γ
6519:∑
6511:−
6503:→
6500:ζ
6422:−
6414:−
6398:−
6366:≥
6359:∑
6343:→
6219:≤
6213:≤
6206:∑
6200:∞
6197:→
6036:ω
6008:ω
5989:ω
5966:…
5907:ω
5892:∑
5868:ω
5848:∞
5828:ω
5765:α
5756:Γ
5749:α
5725:∑
5717:−
5707:∞
5698:∫
5612:∑
5604:−
5594:∞
5585:∫
5500:μ
5481:∫
5443:⋯
5429:μ
5390:μ
5310:μ
5293:∫
5281:μ
5186:⋯
5133:−
5120:∈
5113:∑
5105:π
5079:∞
5073:→
5014:tends to
4944:∑
4912:⋯
4877:∑
4868:∞
4865:→
4746:−
4740:⋯
4683:∞
4668:∑
4655:−
4649:ζ
4567:) is the
4472:⋯
4331:⋯
4125:−
4105:−
4045:−
4023:−
4012:≥
4005:∑
3998:…
3975:Γ
3951:Γ
3931:⋯
3922:−
3910:−
3898:−
3886:−
3874:−
3826:−
3792:−
3781:≥
3774:∑
3717:⋯
3708:−
3690:−
3672:−
3591:⋯
3582:−
3570:−
3517:…
3514:−
3479:−
3453:−
3436:−
3425:…
3403:−
3389:⋯
3355:−
3337:−
3300:≤
3142:⋯
3128:−
3099:−
2949:∞
2934:∑
2926:−
2918:→
2796:∞
2781:∑
2766:−
2746:∞
2731:∑
2597:→
2572:λ
2514:λ
2510:−
2490:∞
2475:∑
2277:−
2266:−
2143:⋯
2089:⋯
2068:−
1988:→
1969:⋯
1769:−
1656:∞
1641:∑
1589:∞
1574:∑
1537:∞
1522:∑
1376:stability
1372:stability
1297:∈
1239:→
1219:bijection
1043:Stability
937:Linearity
683:−
644:−
612:−
547:…
537:≈
509:−
497:∞
488:∫
316:averaging
282:⋯
273:−
261:−
187:∞
172:∑
165:⋯
8646:Category
8412:Absolute
8059:(1949),
8046:(1991),
7963:, p. 21.
7785:See also
2410:Suppose
1905:Suppose
1887:+ ... +
1839:+ ... +
1443:stronger
1335:for all
1282:for all
1157:for all
393:Examples
244:assigns
75:sequence
8422:Uniform
8132:, 2001
8008:, 4.17.
7993:, 4.11.
7864:0515185
6157:
6133:
6129:
6105:
6096:,
4628:
4616:
4597:
4585:
4190:
4175:
3168:), the
2704:. Here
2188:Nørlund
1343:, then
1161:, then
1093:, then
894:regular
751:. Here
720:regular
339:History
329:physics
311:
299:
93:is the
77:of the
8374:Series
8181:series
7946:
7862:
7643:where
6736:(0) +
5544:
5237:, and
5217:where
4796:where
2836:where
1209:. If
1049:). If
945:linear
777:linear
347:define
65:is an
8317:array
8197:Basic
7944:S2CID
7926:arXiv
7797:Notes
5556:then
4803:is a
3321:then
3223:)) =
3180:) as
3160:Then
2872:) as
2852:) as
2384:then
2378:>
1877:+ ...
1452:like
540:0.596
236:is a
83:limit
8257:list
8179:and
7135:<
5225:and
3925:9724
3919:2860
3762:'s.
3306:<
3261:If Σ
3039:log(
3002:) =
2864:for
2436:,...
2327:) =
2190:mean
1478:the
1476:≠ 1,
1456:and
1423:) =
1406:are
1402:and
1390:and
1352:) =
1213:and
1170:) =
1118:) =
1060:and
974:) +
963:) =
928:) =
913:) =
813:and
733:for
61:, a
7936:doi
7698:lim
7481:lim
7343:lim
7331:if
7199:lim
7049:If
6946:sin
6907:lim
6895:if
6826:sin
6783:lim
6771:if
6634:lim
6622:if
6496:lim
6484:if
6336:lim
6324:if
6320:to
6190:lim
6178:if
5978:is
5528:.)
5066:lim
4858:lim
4815:If
4353:If
4346:at
3913:858
3907:264
3191:If
3025:If
2911:lim
2685:If
2590:lim
2446:≥ 0
2415:= {
2353:≥ 0
2219:by
2204:).
2181:as
1360:′).
1185:′).
1133:′).
943:is
856:in
717:is
544:347
232:or
57:In
8663::
8128:,
8122:,
8110:,
8104:,
8091:,
8085:,
7998:^
7983:^
7968:^
7942:.
7934:.
7920:.
7860:MR
7856:19
7854:,
7831:.
7813:.
6131:,
6059:.
5803:.
5546:dx
5538:dμ
5468:dμ
5267:dμ
5042:.
4807:.
4631:,
4594:12
4571:,
4178:−1
3901:84
3895:28
3889:10
3188:.
3035:=
2993::
2695:=
2664:.
2429:,
2422:,
2402:.
2346:).
1991:0.
1870:+
1445:.
1431:).
1398::
1324:=
1266:=
1177:+
1154:+1
1146:=
1125:+
1085:−
1083:+1
1075:=
1017:−
1015:+1
1007:=
1005:+1
988:,
959:+
939:.
904:,
860:.
844:,
829:.
809:,
691:12
351:is
335:.
225:.
85:.
8259:)
8255:(
8169:e
8162:t
8155:v
8115:.
8096:.
8077:.
8067:.
8052:.
8038:.
8028:.
7950:.
7938::
7928::
7922:2
7905:.
7879:.
7848:n
7817:.
7769:.
7766:s
7763:=
7756:2
7752:n
7741:e
7735:n
7731:c
7725:n
7713:+
7709:0
7660:)
7657:x
7654:(
7628:,
7625:s
7622:=
7618:]
7611:+
7605:)
7602:2
7599:+
7596:m
7593:(
7590:)
7587:1
7584:+
7581:m
7578:(
7573:)
7570:1
7564:m
7561:(
7558:m
7550:2
7546:a
7542:+
7536:1
7533:+
7530:m
7526:m
7519:1
7515:a
7511:+
7506:0
7502:a
7497:[
7485:m
7477:=
7471:)
7468:k
7465:+
7462:1
7459:+
7456:m
7453:(
7446:)
7443:k
7437:1
7434:+
7431:m
7428:(
7418:2
7414:]
7410:)
7407:1
7404:+
7401:m
7398:(
7392:[
7384:k
7380:a
7374:m
7369:0
7366:=
7363:k
7347:m
7329:s
7325:1
7322:a
7300:.
7297:x
7294:d
7288:1
7278:)
7274:x
7265:(
7262:)
7259:x
7256:(
7247:A
7236:0
7185:0
7182:a
7178:κ
7176:,
7174:λ
7155:1
7152:+
7149:n
7138:x
7130:n
7115:n
7111:a
7107:+
7101:+
7096:0
7092:a
7088:=
7085:)
7082:x
7079:(
7070:A
7055:n
7051:λ
7023:.
7020:s
7017:=
7014:)
7009:n
7005:a
7001:+
6995:+
6990:1
6986:a
6982:(
6976:h
6971:2
6967:n
6961:h
6958:n
6950:2
6937:n
6924:2
6917:0
6911:h
6893:s
6889:2
6885:1
6882:a
6865:.
6862:s
6859:=
6854:k
6849:)
6843:h
6840:n
6835:h
6832:n
6820:(
6813:n
6809:a
6803:n
6793:0
6787:h
6769:s
6765:k
6761:1
6758:a
6746:f
6742:f
6738:f
6734:f
6702:.
6699:s
6696:=
6690:)
6687:n
6681:+
6678:1
6675:(
6666:n
6662:a
6654:n
6644:0
6620:s
6616:0
6613:a
6591:.
6588:s
6585:=
6580:n
6576:a
6569:)
6566:n
6563:+
6560:1
6557:(
6549:)
6546:n
6540:+
6537:1
6534:(
6523:n
6507:1
6482:s
6478:0
6475:a
6463:k
6459:k
6442:.
6439:s
6436:=
6428:y
6425:n
6418:e
6411:1
6404:y
6401:n
6394:e
6390:y
6387:n
6379:n
6375:a
6369:1
6363:n
6351:+
6347:0
6340:y
6322:s
6314:1
6311:a
6299:δ
6295:δ
6291:δ
6271:.
6268:s
6265:=
6261:]
6256:n
6253:x
6248:[
6242:x
6239:n
6232:n
6228:a
6222:x
6216:n
6210:1
6194:x
6176:s
6172:1
6169:a
6154:2
6151:/
6147:2
6144:s
6140:1
6137:s
6126:2
6123:/
6119:1
6116:s
6112:0
6109:s
6101:1
6098:s
6094:0
6091:s
6045:2
6040:2
6011:2
6003:+
5998:2
5993:2
5963:+
5960:3
5957:+
5954:2
5951:+
5948:1
5921:)
5918:x
5915:(
5912:f
5902:1
5899:=
5896:x
5801:t
5784:t
5781:d
5774:)
5771:1
5768:+
5762:n
5759:(
5746:n
5742:t
5736:n
5732:a
5720:t
5713:e
5702:0
5684:0
5681:a
5677:α
5673:α
5656:.
5653:t
5650:d
5643:!
5640:n
5633:n
5629:t
5623:n
5619:a
5607:t
5600:e
5589:0
5567:n
5562:n
5558:μ
5554:x
5550:x
5542:e
5526:μ
5521:n
5517:μ
5497:d
5493:)
5490:x
5487:(
5484:a
5464:μ
5460:x
5440:+
5433:1
5422:1
5418:x
5412:1
5408:a
5401:+
5394:0
5383:0
5379:x
5373:0
5369:a
5362:=
5359:)
5356:x
5353:(
5350:a
5337:1
5334:a
5330:0
5327:a
5307:d
5301:n
5297:x
5290:=
5285:n
5255:h
5250:h
5246:a
5242:0
5239:a
5235:e
5231:n
5229:(
5227:c
5223:G
5219:H
5202:)
5197:h
5193:a
5189:+
5183:+
5178:0
5174:a
5170:(
5165:)
5162:n
5159:(
5156:H
5151:2
5147:h
5141:2
5138:1
5129:e
5123:Z
5117:h
5102:2
5097:)
5094:n
5091:(
5088:H
5076:+
5070:n
5052:J
5036:e
5032:x
5030:(
5028:J
5016:r
5012:x
5008:r
5004:x
5000:r
4996:J
4976:,
4968:n
4964:x
4958:n
4954:p
4948:n
4936:n
4932:x
4928:)
4923:n
4919:a
4915:+
4909:+
4904:0
4900:a
4896:(
4891:n
4887:p
4881:n
4862:x
4844:0
4841:a
4837:J
4833:x
4829:n
4825:p
4821:x
4819:(
4817:J
4800:k
4798:B
4781:,
4774:1
4771:+
4768:s
4762:1
4759:+
4756:s
4752:B
4743:=
4737:+
4732:s
4728:3
4724:+
4719:s
4715:2
4711:+
4706:s
4702:1
4698:=
4693:s
4689:n
4678:1
4675:=
4672:n
4664:=
4661:)
4658:s
4652:(
4634:ζ
4625:2
4622:/
4619:1
4612:ζ
4607:s
4591:/
4588:1
4581:s
4577:s
4575:(
4573:ζ
4565:s
4563:(
4561:f
4557:A
4553:A
4549:s
4547:(
4545:f
4541:A
4536:i
4532:a
4528:2
4525:a
4521:1
4518:a
4510:s
4506:s
4498:s
4493:n
4489:a
4469:+
4462:s
4457:3
4453:a
4449:1
4444:+
4437:s
4432:2
4428:a
4424:1
4419:+
4412:s
4407:1
4403:a
4399:1
4394:=
4391:)
4388:s
4385:(
4382:f
4355:s
4348:s
4328:+
4321:s
4317:3
4311:3
4307:a
4301:+
4294:s
4290:2
4284:2
4280:a
4274:+
4267:s
4263:1
4257:1
4253:a
4247:=
4244:)
4241:s
4238:(
4235:f
4214:z
4206:n
4202:a
4198:q
4194:q
4185:q
4181:/
4171:z
4144:.
4139:5
4134:=
4131:)
4128:4
4122:;
4119:;
4116:2
4112:/
4108:1
4102:(
4097:0
4093:F
4087:1
4080:=
4074:!
4071:k
4064:k
4060:)
4056:2
4052:/
4048:1
4042:(
4034:k
4030:)
4026:4
4020:(
4015:0
4009:k
4001:=
3928:+
3916:+
3904:+
3892:+
3883:4
3880:+
3877:2
3871:2
3868:+
3865:1
3862:=
3856:)
3851:k
3847:k
3844:2
3838:(
3829:1
3823:k
3820:2
3816:1
3809:1
3806:+
3803:k
3799:)
3795:1
3789:(
3784:0
3778:k
3750:0
3728:3
3725:1
3720:=
3714:+
3711:1
3705:1
3702:+
3699:0
3696:+
3693:1
3687:1
3684:+
3681:0
3678:+
3675:1
3669:1
3647:3
3644:=
3641:n
3638:,
3635:1
3632:=
3629:m
3607:.
3602:n
3599:m
3594:=
3588:+
3585:1
3579:1
3576:+
3573:1
3567:1
3545:1
3542:=
3539:x
3509:n
3506:2
3502:x
3498:+
3493:m
3490:+
3487:n
3483:x
3474:n
3470:x
3466:+
3461:m
3457:x
3450:1
3447:=
3439:1
3433:n
3429:x
3422:+
3419:x
3416:+
3413:1
3406:1
3400:m
3396:x
3392:+
3386:+
3383:x
3380:+
3377:1
3371:=
3363:n
3359:x
3352:1
3345:m
3341:x
3334:1
3309:n
3303:m
3297:1
3287:x
3283:x
3279:x
3275:x
3271:x
3267:n
3263:a
3239:z
3237:(
3235:g
3231:)
3229:z
3227:(
3225:g
3221:z
3219:(
3217:G
3215:(
3213:L
3208:z
3206:(
3204:G
3197:z
3195:(
3193:g
3182:x
3178:x
3176:(
3174:f
3166:s
3164:(
3162:L
3145:.
3139:+
3134:x
3131:3
3124:3
3118:3
3114:a
3110:+
3105:x
3102:2
3095:2
3089:2
3085:a
3081:+
3076:1
3072:a
3068:=
3065:)
3062:x
3059:(
3056:f
3043:)
3041:n
3037:n
3032:n
3028:λ
3015:)
3013:s
3011:(
3008:k
3004:C
3000:s
2998:(
2996:A
2974:.
2969:n
2965:z
2959:n
2955:a
2944:0
2941:=
2938:n
2922:1
2915:z
2907:=
2904:)
2901:s
2898:(
2895:A
2882:s
2880:(
2878:A
2874:z
2870:z
2868:(
2866:f
2854:x
2850:x
2848:(
2846:f
2842:x
2838:z
2821:,
2816:n
2812:z
2806:n
2802:a
2791:0
2788:=
2785:n
2777:=
2772:x
2769:n
2762:e
2756:n
2752:a
2741:0
2738:=
2735:n
2727:=
2724:)
2721:x
2718:(
2715:f
2697:n
2692:n
2688:λ
2669:λ
2649:x
2645:x
2641:f
2624:.
2621:)
2618:x
2615:(
2612:f
2605:+
2601:0
2594:x
2586:=
2583:)
2580:s
2577:(
2568:A
2553:λ
2549:A
2542:x
2523:x
2518:n
2506:e
2500:n
2496:a
2485:0
2482:=
2479:n
2471:=
2468:)
2465:x
2462:(
2459:f
2444:0
2441:λ
2434:2
2431:λ
2427:1
2424:λ
2420:0
2417:λ
2413:λ
2399:k
2395:C
2390:h
2386:C
2382:,
2380:k
2376:h
2367:1
2364:C
2360:0
2357:C
2351:k
2344:s
2342:(
2339:)
2336:p
2333:(
2329:N
2325:s
2323:(
2320:k
2316:C
2310:k
2306:C
2286:)
2280:1
2274:k
2269:1
2263:k
2260:+
2257:n
2251:(
2245:=
2240:k
2235:n
2231:p
2217:p
2202:s
2200:(
2197:p
2193:N
2183:n
2178:n
2176:t
2154:m
2150:p
2146:+
2140:+
2135:1
2131:p
2127:+
2122:0
2118:p
2110:m
2106:s
2100:0
2096:p
2092:+
2086:+
2081:1
2077:s
2071:1
2065:m
2061:p
2057:+
2052:0
2048:s
2042:m
2038:p
2031:=
2026:m
2022:t
2008:p
1980:n
1976:p
1972:+
1966:+
1961:1
1957:p
1953:+
1948:0
1944:p
1937:n
1933:p
1917:0
1914:p
1909:n
1907:p
1893:n
1889:a
1885:0
1882:a
1875:1
1872:a
1868:0
1865:a
1849:n
1845:k
1841:a
1836:1
1833:k
1829:a
1807:r
1778:,
1772:r
1766:1
1762:c
1757:=
1750:)
1747:c
1744:,
1741:r
1738:(
1735:G
1720:,
1717:)
1714:c
1711:,
1708:r
1705:(
1702:G
1698:r
1695:+
1692:c
1689:=
1669:k
1665:r
1661:c
1651:0
1648:=
1645:k
1637:r
1634:+
1631:c
1628:=
1608:1
1605:+
1602:k
1598:r
1594:c
1584:0
1581:=
1578:k
1570:+
1567:c
1564:=
1550:k
1546:r
1542:c
1532:0
1529:=
1526:k
1518:=
1511:)
1508:c
1505:,
1502:r
1499:(
1496:G
1474:r
1439:Σ
1435:A
1429:s
1427:(
1425:B
1421:s
1419:(
1417:A
1412:s
1404:B
1400:A
1392:B
1388:A
1368:a
1364:a
1358:a
1356:(
1354:A
1350:a
1348:(
1346:A
1341:N
1337:i
1331:i
1328:′
1326:a
1321:i
1317:a
1301:N
1294:N
1284:i
1279:)
1277:i
1275:(
1273:f
1270:′
1268:a
1263:i
1259:a
1243:N
1235:N
1231::
1228:f
1215:a
1211:a
1183:a
1181:(
1179:A
1175:0
1172:a
1168:a
1166:(
1164:A
1159:n
1152:n
1148:a
1143:n
1140:′
1138:a
1131:s
1129:(
1127:A
1123:0
1120:s
1116:s
1114:(
1112:A
1107:s
1105:(
1103:A
1099:s
1097:(
1095:A
1090:0
1087:s
1081:n
1077:s
1072:n
1069:′
1067:s
1062:s
1058:0
1055:s
1051:s
1037:A
1033:s
1029:a
1023:n
1019:s
1013:n
1009:s
1003:n
999:a
994:k
990:s
986:r
982:)
980:s
978:(
976:A
972:r
970:(
968:A
965:k
961:s
957:r
954:k
952:(
950:A
941:A
932:.
930:x
926:a
924:(
922:A
917:.
915:x
911:s
909:(
907:A
902:x
898:s
882:A
875:A
769:M
765:Σ
761:Σ
757:M
735:M
715:M
688:1
652:2
649:1
615:1
581:3
578:1
534:x
531:d
524:x
521:+
518:1
512:x
505:e
492:0
454:4
451:1
418:2
415:1
308:2
305:/
302:1
279:+
276:1
270:1
267:+
264:1
258:1
202:.
197:n
194:1
182:1
179:=
176:n
168:=
162:+
157:5
154:1
149:+
144:4
141:1
136:+
131:3
128:1
123:+
118:2
115:1
110:+
107:1
38:.
20:)
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