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4106:{\displaystyle {\begin{aligned}P(z)&=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}}{2^{n}\cdot n}}(z^{2^{n-1}}+z^{2^{n-1}+1}+\cdots +z^{2^{n}-1})\\&=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}}{2^{n}\cdot n}}(z^{2^{n-1}}+z^{2^{n-1}+1}+\cdots +z^{2^{n-1}+2^{n-2}+\cdots +2^{1}+1})\\&=\sum _{n=1}^{\infty }{\frac {(-1)^{\log _{2}2^{n-1}}}{2^{n}\cdot (\log _{2}2^{n-1}+1)}}z^{2^{n-1}}+{\frac {(-1)^{\log _{2}2^{n-1}}}{2^{n}\cdot (\log _{2}2^{n-1}+1)}}z^{2^{n-1}+1}+\cdots +{\frac {(-1)^{\log _{2}2^{n-1}}}{2^{n}\cdot (\log _{2}2^{n-1}+1)}}z^{2^{n-1}+2^{n-1}\cdots +2^{1}+1}\\&=\sum _{n=1}^{\infty }{\frac {(-1)^{\lfloor \log _{2}2^{n-1}\rfloor }}{2^{n}\cdot (\lfloor \log _{2}2^{n-1}\rfloor +1)}}z^{2^{n-1}}+{\frac {(-1)^{\lfloor \log _{2}(2^{n-1}+1)\rfloor }}{2^{n}\cdot (\lfloor \log _{2}(2^{n-1}+1)\rfloor +1)}}z^{2^{n-1}+1}+\cdots +{\frac {(-1)^{\lfloor \log _{2}(2^{n-1}+2^{n-2}+\cdots +2^{1}+1)\rfloor }}{2^{n}\cdot (\lfloor \log _{2}(2^{n}+2^{n-1}+\cdots +2^{1}+1)\rfloor +1)}}z^{2^{n-1}+2^{n-1}\cdots +2^{1}+1}\\&=\sum _{n=1}^{\infty }{\frac {(-1)^{\lfloor \log _{2}2^{n-1}\rfloor }}{2^{\lfloor \log _{2}2^{n-1}\rfloor +1}\cdot (\lfloor \log _{2}2^{n-1}\rfloor +1)}}z^{2^{n-1}}+{\frac {(-1)^{\lfloor \log _{2}(2^{n-1}+1)\rfloor }}{2^{\lfloor \log _{2}(2^{n-1}+1)\rfloor +1}\cdot (\lfloor \log _{2}(2^{n-1}+1)\rfloor +1)}}z^{2^{n-1}+1}+\cdots +{\frac {(-1)^{\lfloor \log _{2}(2^{n-1}+2^{n-2}+\cdots +2^{1}+1)\rfloor }}{2^{\lfloor \log _{2}(2^{n-1}+2^{n-2}+\cdots +2^{1}+1)\rfloor +1}\cdot (\lfloor \log _{2}(2^{n}+2^{n-1}+\cdots +2^{1}+1)\rfloor +1)}}z^{2^{n-1}+2^{n-1}\cdots +2^{1}+1}\\&=\sum _{m=1}^{\infty }{\frac {(-1)^{\lfloor \log _{2}m\rfloor }}{2^{\lfloor \log _{2}m\rfloor +1}\cdot (\lfloor \log _{2}m\rfloor +1)}}z^{m}\end{aligned}}}
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SINGULARITY)on the circle of convergence, even though the series may converge everywhere on the circle. "Singular point" doesn't mean a point where the function is undefined, rather I think it means a point from which analytic continuation is impossible. (See Knopp, I'll try to get the page number asap.) For an example where I see no non-removable singularity(which I interpret as failing to converge, that is, "blowing up"), consider (/), summed from n=1 to infinity.
22:
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already "nominally" not a power series. When you add in that it is conditionally convergent on the boundary, you have a series that probably can't even be rearranged to be a power series without changing the values of the function on boundary points. So I think it should be pointed out that it isn't a power series, though closely related. Best wishes,
1572:
collected its terms its so as to have exponents in increasing order. That may be why the authors in the original source wrote it like this--it was constructed to show that a series of functions could converge uniformly but not absolutely on the boundary--and they had been unable(it may even be impossible, afik)to find a power series with that property.
4791:
It is easy to find the radius of convergence but how to determine what happens on the radius of convergence. What is special here is that all of the series mentioned have a radius of convergence of 1. A good idea would also be to include these 3 examples in the relevent section as they are an interesting example due to their 3 properties.
968:
of convergence is 1, so the number 1 is indeed on the boundary, and (according to Knopp) the number 1 is a singular point. So there you have a singular point on the boundary. In order to show that the article's statement is wrong, you'd need to give an example in which there is NO singular point on the boundary.
4486:. This tends to one, since eventually either the top and the bottom are either equal or the bottom is larger than the top by one. The ratio has a factor of β1 which we can ignore since we're interested in the absolute value. Finally, we take the lim sup to conclude that the radius of convergence is one.
952:"Exercise. The unit circle is the circle of convergence of the power series . Show that the point +1 is a singular point of the function represented by the series in the unit circle, by expanding in a new power series with center z1 = +1/2. (Nevertheless, the given series is convergent for z = +1!!)"
1451:
Lots of power series don't converge, so I think you are saying by "it is a theorem" that there is a theorem if a power series does converge at a point, then it converges absolutely. However, in example 4 it says it doesn't converge absolutely on the boundary. If this is true, and if the theorem which
1417:
But I think it's not a power series, since it's not absolutely convergent. If you rearrage the terms so that it's a power series, you risk losing convergence on the boundary or at least changing what it converges to, thus it's a different function than the power series you rearranged it to. Shouldn't
1028:
I'm not sure what you mean by this distinction. Maybe I'll take a look at Knopp and see what he says. I confess I've never seen an example quite like this. At any rate, this function cannot be extended to a holomorphic function defined on any disk centered at 0 whose radius is more than 1, and not
967:
Well, you quote Knopp as saying that 1 is a singular point, and if Knopp is right, then that is not an example that shows the article is wrong, but is in perfect agreement with the article. The article says there must be a singular point on the boundary. In this case, the center is 0 and the radius
926:
You must realize that I'm at the outer limits of my knowledge here. Part of why I think it says they must blow up(that is, diverge) somewhere on the circle is that both examples that are given do so. My mis?understanding of the situation is that there will always be at least one "singular point" (not
357:
I've looked it up, and I can't seem to find any reference to the radius of convergence for series that are not power series. Is radius of convergence only definined for power series? Also, the ratio and root tests given on this page are not the same as the ratio and root tests that this page links to
4790:
Can someone actually do the example of showing and actually working out the steps that the power series Ξ£ z/n converges at every point of the unit circle except at z=1, that the series Ξ£ z/n converges at every point of the unit circle and that Ξ£ nz does not converges on any point of the unit circle.
4131:
That's pretty nice mathematics! I'm sure you're correct that the radius of convergence is 1, and I agree that we want it in the form given in the article, and I think it should be retained in the article. But, because it a series of of polynomials rather than a series of increasing powers of z, it's
1292:
Many apologies to all concerned. I copied over the text of the "theorem" stated in the old version of the article without thinking it through. There is no requirement that there be a non-removable singularity on boundary of the circle of convergence, merely that a point exists where the function can
1276:
No, it is all of the possible singularities. Essential singularities are technically defined as all singularities that are neither poles nor removable, and then their other properties are proved from that definition. In the example given above, since the power series has a radius of convergence of 1
1300:
Power series define a holomorphic function within their domain of convergence, which is always a disk of some radius (possibly null, possibly infinite). Anything can happen on the boundary of the disk -- the series can converge everywhere, diverge everywhere, converge at some points but not others,
863:
I think you mean "to the nearest SINGULAR POINT", not "singularity". It sounds like you're saying by "series MUST be bad at at least one point" that the series must diverge somewhere on the circle of convergence, which I disagree with, see article. Also see Konrad Knopp, Theory of
Functions Part I,
769:
Suppose one had a power series w/radius of conv = 1, say, around z=0, say. If z were allowed to be quaternionic, then the boundary of convergence would be a 3 sphere of rad 1. I bet the subset of the boundary where the series converges could be quite beautiful and fascinating. Does anyone know of
4146:
OK, I still disagree with you on this one. I agree that it's written as a series of polynomials. I also agree that the given expression is not the usual expression for a power series. But I still believe that the series is a power series; it's just had some of its consecutive terms collected. My
1237:. Within the radius of convergence of a power series (not on the boundary, but the interior of the circle), the series converges absolutely, hence it is bounded. Thus, the only possible singularities are of the removable type, and they can be removed, leaving a holomoprhic function behind. Best,
1571:
Right, I was just about to post an addendum to that effect as it had finally occurred to me that if this is the theorem Ozub was referring to, it was not meant to hold on the boundary. But then
Example 4 is quite possibly not equivalent to the power series one would obtain if one rearranged and
1375:
provides such a case. In this case, if we were to take a power series centered about 1, it would have a radius of convergence 1, the series would converge properly to 0 at 0, but we couldn't extend it further. 0 is not a singularity of the square root so defined -- we converge to 0 along every
1309:
From these two theorems we conclude that at least one point on the boundary of a power series' domain of convergence (which is a disc), must be on the boundary of the set where the function can be holomorphically defined. This need not mean a singularity -- there are ways to define holomorphic
4817:
This section states "The limit involved in the ratio test is usually easier to compute, and when that limit exists, it shows that the radius of convergence is finite." Surely this is incorrect? It's possible that the ratio tends to some <1 value irrespective of the relevant variable, no?
335:
Ratio test doesn't have a letter assocaited with it on the page at least (anymore?). But I don't want to change it if people actually use the letters L or C to mean something. If its 100% arbitrarily used.... Well, yea i guess it would be a good parallel. I think i'll fix it up so it is less
1181:. These are equivalent (by a theorem whose proof I do not have at hand) to functions which are bounded in a neighborhood of the singularity. You can "remove" the singularity by defining the function to be its limit at that point; there will be a unique limit - this is a theorem by Riemann.
658:
Oh, thats interesting, I should have seen that. That would be an insightful addition to this page, I think i'll add it. I noticed that "region of convergence" redirects here, while it has no explanation whatsoever. I don't know what I can about it, but something should be added about it.
1228:
about 0 (try approaching 0 along the real line from either direction. Then do the same along the imaginary directions -- you will see that in some directions there is an limit as you approach, in other directions it diverges, but along the imaginary axis it is oscillating but of bounded
1393:
Hah. Changed the article again -- it's not that the function can't be defined at a point on the circle, it's that the function cannot be defined past the circle. Ugh. My apologies to everybody on this project. I hope we'll get there eventually -- please do keep checking my workΒ :)
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1598:
Yes, absolute convergence in the interior is the theorem I'm thinking of. I think the confusion is that example 4 is a power series with its exponents in increasing order, but it's not written in the usual way because the usual way is more complicated. Let me demonstrate:
273:
986:
is linked in the clarity & simplicity section, but I don't see any example in the singularity article that resembles Knopp's "singular point." Perhaps the clarity&simplicty sect. should have an example like Knopp's above. Apart from that, I think
1159:
This makes it clear the radius of convergence must be less than or equal to 1. But to play devil's advocate, why must it equal 1? Couldn't there be singular points, perhaps of the type mentioned above by Konrad Knopp, still closer to the origin?
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about 0, that means there can be no non-removable singularities inside the disc of radius one centered at 0. I don't know how to phrase it more clearly. Can you try rephrasing and explaining your question at length? Best,
625:
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1193:. The function will, in an arbitrarily small neighborhood of the singularity, assume every possible value except one. Infinitely often. This is a highly nontrivial result by Picard. An example of such a function is
527:
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Within any disc (as large as you wish) strictly contained within a domain where a function is holomorphic, the power series representation of the function, taken about the point at the center of the domain, is
908:
It doesn't say "blows up"; it says "has a non-removable singularity". Rich, can you cite an example of a power series with a finite radius of convergence that has no non-removable singularity on the boundary?
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of the circle of convergence, then it converges absolutely at that point. That leaves open the question of a power series converges absolutely at boundary points where it converges. A simple example is
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Absolute convergence is not a necessary condition for being a power series; it is a theorem. The terms are already arranged so that the series is a power series. It's written with several powers of
692:
I disagree. This is an important enough concept to warrant its own article. Besides, the radius of convergence is only for power series of one variable, while power series can be in many variables.
4117:
From this it's pretty straightforward to see that the radius of convergence is one. But in order to really show the facts about the boundary, I think you want it in the form given in the article.
1452:
I am supposing you are citing is correct, then
Example 4 is not a power series on the points of the boundary. But it may be that you meant something slightly different was a theorem. Best wishes,
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no longer be defined holomorphically. This situation may arise in the case of functions with branch cuts, such as the square root function or the logarithm defined over the complex numbers.
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possible curve approaching it. However, you cannot produce a power series of positive radius of convergence, for this definition of the complex square root, in any open neighborhood of 0.
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The complex square root is an excellent example, I think. A pretty standard way to define it is to define it everywhere except along the negative part of the real axis. This example at
4744:
Can please someone do a few more concrete examples of finding out the radius of convergence for when z is complex and when it is real and a few special cases. It will help a lot.
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In fact, the series MUST be bad at at least one point on the circle |z-a|=r since the radius of convergence is equal to the distance from point "a" to the nearest singularity. --
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The difference between this and the current presentation is that it will be more clear that we have to consider convergence of all the partial sums (one for each
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in each summand because this way of writing it avoids some complications with base 2 logs and floor functions. (Also, that's the way the original source has it.)
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that are at least 2, and these terms are already in order. A perhaps more convincing reason is that you can represent the series as a sequence of partial sums
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I may as well include the justification for the radius of convergence being one: Use the ratio test. The ratio has a factor of the form 2 to the power
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This situation may arise in the case of functions with branch cuts, such as the square root function or the logarithm defined over the complex numbers.
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With hopes of ending the confusion over what's been happening, I will give a more formal summary. There are two theorems that are relevant here:
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That's a limiting case. Many things can happen depending on the function. You may have divergence, or convergence, or conditional convergence.
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many series have no point on circle of conv where it blows up. But that is what i think the section is saying. I put expert needed tag on it.
1350:
This need not mean a singularity -- there are ways to define holomorphic functions on domains that leave out a good deal more than a point.
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It seems right to me. Where do you think the section says that all series have a point on the circle of convergence where they blow up? --
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4674:{\displaystyle \sum _{i=1}^{\infty }a_{i}z^{i}{\text{ where }}a_{i}={\frac {(-1)^{n-1}}{2^{n}n}}{\text{ for }}2^{n-1}\leq i<2^{n}.}
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948:
Theory of
Functions by Konrad Knopp, Part I, translated by Frederick Bagemihl. 1945, Dover. Pages 99-104 are relevant. On page 104:
1310:
functions on domains that leave out a good deal more than a point. I hope this clears up the confusion over this article. Best,
638:, the region of convergence will most likely not be a circle, and thus it makes no sense to talk of a radius of convergence. --
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268:{\displaystyle {1 \over L}=R={\mbox{Radius Of Convergence}}=\lim _{n\rightarrow \infty }\left|{\frac {c_{n}}{c_{n}+1}}\right|}
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Perhaps you're right, but I think attention should be drawn to the distinction between a singular point and a singularity.
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Why must 1/L and 1/C be used for the radius of convergence? Why can't we invert the equations as they stand? For example
358:(I think), because here, the root test is applied to only the coeffecients of the term of the power series, while on the
44:
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of convergence, or even polyballs("Hartogs stuff') or more general cartesian products of balls of different dimension.
682:, and that page already has something on radius of convergence, it seems only natural to merge the two. Any comments?
377:
731:" - That doesn't make sense. A test is not an equation, nor is it a real number. Also for "ratio test" later. --
4226:. So these two sequences of partial sums have the same limit, and hence their behavior at the boundary is the same.
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In addition, maybe we should start a new article on "singular point" in Knopp's sense, or put it in a section of
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1060:. Perhaps it would help if these three examples include an explanation of the singularities involved? --
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That may be all of the "singularities" but it's not all of the singular (nonremovable)points.Regards,
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Alright that it, I'm freaking changing it. If someone wants to change it back, discuss it *here*.
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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A vocabulary point -- there are three types of possible singularities in the complex plane.
1104:"... However, we could also have found the radius of convergence by noting that the function
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1187:. The function will "blow up" to infinity along every curve of approach to the singularity.
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actually asserted that the root test is a number. I've done some editing on that one too.
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tends to zero; so 2 raised to this quantity is one or 1/2. The ratio also has a factor of
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4164:. If you call the partial sums of the power series at the bottom of the above derivation
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should be applied to the whole term, so the root test says that the series converges if
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Does that mean you don't consider a branch point to be a non-removable singularity?
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results/research in this area? It would be a great addition to put in this article.
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4479:{\displaystyle (\lfloor \log _{2}m\rfloor +1)/(\lfloor \log _{2}(m+1)\rfloor +1)}
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and humility support keeping the experts needed tag. What do you think?Regards,
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I'm now convinced that you are correct. Also, the suggestion by Zero is good.
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What would you like examples of? What sorts of series? (Please keep in mind
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620:{\displaystyle |z|<{\frac {1}{\limsup _{n\to \infty }{\sqrt{|c_{k}|}}}},}
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Well, you have to do a bit of work. A power series is a series of the form
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I don't understand your objection, but I believe the article is correct.
4320:{\displaystyle \lfloor \log _{2}m\rfloor -\lfloor \log _{2}(m+1)\rfloor }
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I also agree. I've done some cleanup accordingly. The article titled
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I bet in power series of many variables one has many radii to define a
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Since it looks as if a "radius of convergence" is a term only used for
4331:
sufficiently large, this quantity is either zero or minus one because
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this example be labelled as a more general object than power series?
933:
Even if I'm wrong about this, hopefully I'll learn something. Regards
4511:
I suggest that in the article the example be displayed like this:
1469:
The theorem is that if a power series converges at a point in the
741:
I agree. I have no idea what is being said. Can someone fix this?
522:{\displaystyle \limsup _{n\to \infty }{\sqrt{|c_{k}z^{k}|}}<1.}
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Radius of convergence#Convergence on the "circle of convergence"
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Can you cite examples of the sort of thing you have in mind?
275:. Why can't we put that up. Do C and L have special uses?? -
864:
Dover, page 100, and
Exercise 2 on page 73. --Rich Peterson
1074:
I see that the article you linked to doesn't mention the
1056:. This article has a very similar example, example #3 at
1528:{\displaystyle \sum _{n=1}^{\infty }{\frac {z^{n}}{n}}.}
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merely because of a failure of the series to converge.
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point is that no term rearrangement is necessary: The
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1409:Example 4 in Convergence on the Boundary
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1098:The current version, on 7/15/2008, has:
60:
19:
1048:. This function has a branch point at
631:which is the formulation on this page.
301:A possible reason is consistency with
4883:B-Class vital articles in Mathematics
1156:, which are at a distance 1 from 0."
353:Radius of convergence of other series
7:
1548:=Β 1. It converges conditionally at
106:This article is within the scope of
49:It is of interest to the following
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729:The root test is defined as: C=...
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362:page, the entire term is applied.
220:
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4893:Mid-priority mathematics articles
4219:{\displaystyle S_{p}=T_{2^{p}-1}}
309:. But feel free to change it. --
126:Knowledge:WikiProject Mathematics
4863:Knowledge level-5 vital articles
4377:{\displaystyle \log _{2}m/(m+1)}
1235:there are no other singularities
533:This condition is equivalent to
129:Template:WikiProject Mathematics
93:
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1540:This converges absolutely for |
146:This article has been rated as
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833:17:38, 11 February 2007 (UTC)
120:and see a list of open tasks.
4888:B-Class mathematics articles
4807:07:38, 11 October 2010 (UTC)
1413:a series of functions, but--
1318:) 04:17, 18 July 2008 (UTC)
760:21:54, 31 October 2006 (UTC)
746:21:06, 31 October 2006 (UTC)
715:21:33, 31 October 2006 (UTC)
169:L and C: root and ratio test
4813:Theoretical radius - error?
4786:Convergence on the boundry.
4780:22:15, 4 October 2010 (UTC)
4760:16:07, 4 October 2010 (UTC)
1260:) 03:41, 18 July 2008 (UTC)
736:08:59, 14 August 2006 (UTC)
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4843:16:40, 20 March 2016 (UTC)
4828:10:56, 20 March 2016 (UTC)
4730:16:26, 21 March 2010 (UTC)
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4142:22:18, 17 March 2010 (UTC)
4127:23:02, 16 March 2010 (UTC)
1582:20:52, 16 March 2010 (UTC)
1562:20:04, 16 March 2010 (UTC)
1544:|Β <Β 1. It diverges at
1462:19:37, 16 March 2010 (UTC)
1447:11:02, 16 March 2010 (UTC)
1428:08:28, 16 March 2010 (UTC)
702:17:42, 29 March 2006 (UTC)
687:10:29, 29 March 2006 (UTC)
664:10:19, 29 March 2006 (UTC)
648:07:29, 29 March 2006 (UTC)
367:01:21, 29 March 2006 (UTC)
341:09:23, 28 March 2006 (UTC)
319:12:03, 27 March 2006 (UTC)
289:08:25, 27 March 2006 (UTC)
4704:An excellent idea! Done!
1404:04:43, 18 July 2008 (UTC)
1386:04:34, 18 July 2008 (UTC)
1367:04:27, 18 July 2008 (UTC)
1328:04:17, 18 July 2008 (UTC)
1287:04:02, 18 July 2008 (UTC)
1270:03:42, 18 July 2008 (UTC)
1247:17:43, 17 July 2008 (UTC)
1170:18:45, 15 July 2008 (UTC)
1088:16:45, 27 June 2008 (UTC)
1070:12:41, 27 June 2008 (UTC)
1039:02:51, 27 June 2008 (UTC)
1019:00:51, 27 June 2008 (UTC)
1001:00:43, 27 June 2008 (UTC)
978:20:17, 26 June 2008 (UTC)
962:20:06, 25 June 2008 (UTC)
943:19:13, 25 June 2008 (UTC)
919:14:47, 24 June 2008 (UTC)
903:13:19, 24 June 2008 (UTC)
888:03:40, 24 June 2008 (UTC)
821:{\displaystyle |z-a|=r\!}
145:
78:
57:
765:3-sphere of convergence?
152:project's priority scale
1221:{\displaystyle e^{1/z}}
1191:Essential Singularities
1179:Removable singularities
859:18:52, 1 May 2007 (UTC)
674:Merge with power series
109:WikiProject Mathematics
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3269:
3266:
3261:
3258:
3255:
3250:
3247:
3244:
3240:
3236:
3231:
3227:
3223:
3219:
3211:
3206:
3203:
3200:
3196:
3192:
3187:
3183:
3179:
3175:
3171:
3168:
3165:
3157:
3152:
3149:
3146:
3142:
3138:
3135:
3133:
3131:
3130:
3125:
3122:
3117:
3113:
3109:
3106:
3101:
3098:
3095:
3091:
3087:
3082:
3079:
3076:
3072:
3067:
3060:
3057:
3054:
3051:
3048:
3045:
3042:
3037:
3033:
3029:
3026:
3023:
3018:
3015:
3012:
3008:
3004:
2999:
2995:
2991:
2988:
2983:
2979:
2975:
2972:
2969:
2964:
2960:
2952:
2949:
2946:
2943:
2938:
2934:
2930:
2927:
2924:
2919:
2916:
2913:
2909:
2905:
2900:
2897:
2894:
2890:
2886:
2883:
2878:
2874:
2870:
2866:
2862:
2859:
2856:
2850:
2847:
2844:
2839:
2836:
2831:
2828:
2825:
2821:
2816:
2809:
2806:
2803:
2800:
2797:
2794:
2791:
2786:
2783:
2780:
2776:
2772:
2769:
2764:
2760:
2756:
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2741:
2733:
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2724:
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2709:
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2697:
2693:
2689:
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2669:
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2659:
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2652:
2647:
2640:
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2634:
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2623:
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2616:
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2607:
2603:
2599:
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2593:
2588:
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2571:
2568:
2565:
2561:
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2552:
2548:
2544:
2540:
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2533:
2530:
2522:
2517:
2514:
2511:
2507:
2503:
2500:
2498:
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2318:
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2299:
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2269:
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2263:
2259:
2255:
2250:
2246:
2242:
2239:
2234:
2230:
2220:
2217:
2214:
2210:
2206:
2201:
2197:
2192:
2188:
2185:
2182:
2176:
2169:
2166:
2163:
2159:
2154:
2147:
2144:
2141:
2136:
2133:
2130:
2126:
2122:
2117:
2113:
2109:
2106:
2101:
2097:
2087:
2084:
2081:
2077:
2073:
2068:
2064:
2059:
2055:
2052:
2049:
2041:
2036:
2033:
2030:
2026:
2022:
2019:
2017:
2015:
2014:
2011:
2006:
2003:
1998:
1994:
1990:
1987:
1984:
1979:
1976:
1973:
1969:
1965:
1960:
1957:
1954:
1950:
1945:
1941:
1938:
1935:
1930:
1927:
1922:
1919:
1916:
1912:
1907:
1903:
1896:
1893:
1890:
1886:
1881:
1877:
1871:
1868:
1863:
1859:
1851:
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1841:
1837:
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1818:
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1808:
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1801:
1799:
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1671:
1667:
1663:
1660:
1657:
1649:
1644:
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1638:
1634:
1630:
1627:
1625:
1623:
1620:
1617:
1614:
1611:
1610:
1589:
1588:
1587:
1586:
1585:
1584:
1538:
1537:
1535:
1524:
1519:
1514:
1510:
1502:
1497:
1494:
1491:
1487:
1467:
1466:
1465:
1464:
1414:
1407:
1391:
1390:
1389:
1388:
1355:
1353:
1352:
1351:
1346:
1343:
1341:
1340:
1339:
1334:
1307:
1306:
1302:
1290:
1289:
1273:
1272:
1231:
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1215:
1211:
1207:
1203:
1188:
1182:
1145:
1142:
1122:
1119:
1116:
1113:
1102:
1101:
1100:
1099:
1093:
1092:
1091:
1090:
1026:
1025:
1024:
1023:
1022:
1021:
950:
949:
931:
930:
929:
928:
906:
905:
875:
872:
870:
868:
867:
866:
865:
851:
850:
815:
812:
808:
804:
801:
798:
794:
781:
778:
766:
763:
749:
748:
724:
721:
720:
719:
718:
717:
675:
672:
671:
670:
669:
668:
667:
666:
651:
650:
632:
629:
628:
627:
616:
607:
601:
595:
591:
586:
577:
574:
571:
567:
562:
557:
553:
549:
545:
531:
530:
529:
518:
515:
509:
503:
497:
493:
487:
483:
478:
469:
466:
463:
459:
437:
436:
435:
424:
419:
415:
409:
405:
399:
394:
391:
388:
384:
354:
351:
350:
349:
348:
347:
346:
345:
344:
343:
326:
325:
324:
323:
322:
321:
294:
293:
292:
291:
263:
257:
254:
249:
245:
238:
234:
228:
222:
219:
216:
212:
208:
198:
195:
192:
187:
184:
170:
167:
164:
163:
160:
159:
156:
155:
144:
138:
137:
135:
118:the discussion
105:
104:
88:
76:
75:
67:
55:
54:
48:
26:
13:
10:
9:
6:
4:
3:
2:
4905:
4894:
4891:
4889:
4886:
4884:
4881:
4879:
4876:
4874:
4871:
4869:
4866:
4864:
4861:
4859:
4856:
4855:
4853:
4844:
4840:
4836:
4832:
4831:
4830:
4829:
4825:
4821:
4820:188.74.64.241
4812:
4810:
4808:
4804:
4800:
4796:
4785:
4781:
4777:
4773:
4769:
4765:
4764:
4763:
4761:
4757:
4753:
4749:
4740:More examples
4739:
4731:
4727:
4723:
4719:
4718:
4717:
4716:
4715:
4711:
4707:
4703:
4702:
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4697:
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4689:
4668:
4663:
4659:
4655:
4652:
4649:
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4638:
4634:
4622:
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4605:
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4599:
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4579:
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4570:
4559:
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4545:
4534:
4531:
4528:
4524:
4516:
4515:
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4512:
4497:
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4470:
4467:
4458:
4455:
4452:
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4441:
4437:
4426:
4419:
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4410:
4407:
4402:
4398:
4368:
4365:
4362:
4355:
4351:
4348:
4343:
4339:
4330:
4308:
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4302:
4296:
4291:
4287:
4280:
4274:
4271:
4266:
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4250:
4249:
4248:
4247:
4246:
4245:
4244:
4243:
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4211:
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4199:
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4190:
4185:
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4167:
4162:
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4145:
4144:
4143:
4139:
4135:
4130:
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4124:
4120:
4116:
4094:
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4080:
4077:
4071:
4068:
4063:
4059:
4049:
4044:
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4035:
4032:
4027:
4023:
4015:
4004:
4001:
3996:
3992:
3980:
3977:
3961:
3958:
3955:
3951:
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3900:
3896:
3891:
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3876:
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3854:
3851:
3846:
3842:
3838:
3835:
3832:
3827:
3824:
3821:
3817:
3813:
3808:
3804:
3797:
3792:
3788:
3778:
3773:
3770:
3761:
3758:
3753:
3749:
3745:
3742:
3739:
3734:
3731:
3728:
3724:
3720:
3715:
3712:
3709:
3705:
3698:
3693:
3689:
3681:
3667:
3664:
3659:
3655:
3651:
3648:
3645:
3640:
3637:
3634:
3630:
3626:
3621:
3618:
3615:
3611:
3604:
3599:
3595:
3583:
3580:
3571:
3568:
3565:
3560:
3557:
3552:
3549:
3546:
3542:
3537:
3527:
3524:
3515:
3512:
3507:
3504:
3501:
3497:
3490:
3485:
3481:
3471:
3466:
3463:
3454:
3451:
3446:
3443:
3440:
3436:
3429:
3424:
3420:
3412:
3398:
3395:
3390:
3387:
3384:
3380:
3373:
3368:
3364:
3352:
3349:
3340:
3333:
3330:
3327:
3323:
3318:
3308:
3305:
3297:
3294:
3291:
3287:
3283:
3278:
3274:
3264:
3259:
3256:
3248:
3245:
3242:
3238:
3234:
3229:
3225:
3217:
3204:
3201:
3198:
3194:
3190:
3185:
3181:
3169:
3166:
3150:
3147:
3144:
3140:
3136:
3134:
3123:
3120:
3115:
3111:
3107:
3104:
3099:
3096:
3093:
3089:
3085:
3080:
3077:
3074:
3070:
3065:
3055:
3052:
3043:
3040:
3035:
3031:
3027:
3024:
3021:
3016:
3013:
3010:
3006:
3002:
2997:
2993:
2986:
2981:
2977:
2967:
2962:
2958:
2944:
2941:
2936:
2932:
2928:
2925:
2922:
2917:
2914:
2911:
2907:
2903:
2898:
2895:
2892:
2888:
2881:
2876:
2872:
2860:
2857:
2848:
2845:
2842:
2837:
2834:
2829:
2826:
2823:
2819:
2814:
2804:
2801:
2792:
2789:
2784:
2781:
2778:
2774:
2767:
2762:
2758:
2748:
2743:
2739:
2725:
2722:
2717:
2714:
2711:
2707:
2700:
2695:
2691:
2679:
2676:
2667:
2660:
2657:
2654:
2650:
2645:
2635:
2632:
2624:
2621:
2618:
2614:
2610:
2605:
2601:
2591:
2586:
2582:
2569:
2566:
2563:
2559:
2555:
2550:
2546:
2534:
2531:
2515:
2512:
2509:
2505:
2501:
2499:
2488:
2485:
2480:
2476:
2472:
2469:
2464:
2461:
2458:
2454:
2450:
2445:
2442:
2439:
2435:
2430:
2420:
2417:
2412:
2409:
2406:
2402:
2398:
2393:
2389:
2382:
2377:
2373:
2363:
2360:
2357:
2353:
2349:
2344:
2340:
2331:
2328:
2319:
2316:
2313:
2308:
2305:
2300:
2297:
2294:
2290:
2285:
2275:
2272:
2267:
2264:
2261:
2257:
2253:
2248:
2244:
2237:
2232:
2228:
2218:
2215:
2212:
2208:
2204:
2199:
2195:
2186:
2183:
2174:
2167:
2164:
2161:
2157:
2152:
2142:
2139:
2134:
2131:
2128:
2124:
2120:
2115:
2111:
2104:
2099:
2095:
2085:
2082:
2079:
2075:
2071:
2066:
2062:
2053:
2050:
2034:
2031:
2028:
2024:
2020:
2018:
2004:
2001:
1996:
1992:
1988:
1985:
1982:
1977:
1974:
1971:
1967:
1963:
1958:
1955:
1952:
1948:
1943:
1939:
1936:
1933:
1928:
1925:
1920:
1917:
1914:
1910:
1905:
1901:
1894:
1891:
1888:
1884:
1879:
1869:
1866:
1861:
1857:
1849:
1846:
1843:
1835:
1832:
1816:
1813:
1810:
1806:
1802:
1800:
1786:
1783:
1778:
1774:
1769:
1765:
1762:
1759:
1754:
1751:
1746:
1743:
1740:
1736:
1731:
1727:
1720:
1717:
1714:
1710:
1705:
1695:
1692:
1687:
1683:
1675:
1672:
1669:
1661:
1658:
1642:
1639:
1636:
1632:
1628:
1626:
1618:
1612:
1601:
1600:
1597:
1596:
1595:
1594:
1593:
1592:
1591:
1590:
1583:
1579:
1575:
1570:
1569:
1568:
1567:
1566:
1565:
1564:
1563:
1559:
1555:
1554:Michael Hardy
1551:
1547:
1543:
1536:
1522:
1517:
1512:
1508:
1495:
1492:
1489:
1485:
1477:
1476:
1475:
1472:
1463:
1459:
1455:
1450:
1449:
1448:
1444:
1440:
1436:
1432:
1431:
1430:
1429:
1425:
1421:
1412:
1408:
1406:
1405:
1401:
1397:
1387:
1383:
1379:
1374:
1370:
1369:
1368:
1364:
1360:
1359:Michael Hardy
1356:
1354:
1349:
1348:
1347:
1344:
1342:
1337:
1336:
1335:
1332:
1331:
1330:
1329:
1325:
1321:
1317:
1313:
1303:
1299:
1298:
1297:
1294:
1288:
1284:
1280:
1275:
1274:
1271:
1267:
1263:
1259:
1255:
1251:
1250:
1249:
1248:
1244:
1240:
1236:
1213:
1209:
1205:
1201:
1192:
1189:
1186:
1183:
1180:
1177:
1176:
1175:
1172:
1171:
1167:
1163:
1157:
1143:
1140:
1117:
1111:
1097:
1096:
1095:
1094:
1089:
1085:
1081:
1080:Michael Hardy
1077:
1073:
1072:
1071:
1067:
1063:
1059:
1055:
1051:
1047:
1043:
1042:
1041:
1040:
1036:
1032:
1031:Michael Hardy
1020:
1016:
1012:
1008:
1004:
1003:
1002:
998:
994:
990:
985:
981:
980:
979:
975:
971:
970:Michael Hardy
966:
965:
964:
963:
959:
955:
947:
946:
945:
944:
940:
936:
925:
924:
923:
922:
921:
920:
916:
912:
911:Michael Hardy
904:
900:
896:
892:
891:
890:
889:
885:
881:
873:
871:
862:
861:
860:
857:
853:
852:
849:
845:
841:
837:
836:
835:
834:
831:
813:
810:
802:
799:
796:
779:
777:
776:
773:
764:
762:
761:
758:
757:Michael Hardy
754:
747:
744:
740:
739:
738:
737:
734:
730:
722:
716:
713:
709:
705:
704:
703:
699:
695:
691:
690:
689:
688:
685:
681:
673:
665:
662:
657:
656:
655:
654:
653:
652:
649:
645:
641:
637:
633:
630:
614:
605:
593:
589:
569:
560:
555:
547:
535:
534:
532:
516:
513:
507:
495:
491:
485:
481:
461:
449:
448:
446:
442:
438:
422:
417:
413:
407:
403:
392:
389:
386:
382:
374:
373:
371:
370:
369:
368:
365:
361:
352:
342:
339:
334:
333:
332:
331:
330:
329:
328:
327:
320:
316:
312:
308:
304:
300:
299:
298:
297:
296:
295:
290:
287:
283:
282:
281:
280:
279:
278:
261:
255:
252:
247:
243:
236:
232:
226:
214:
206:
196:
193:
190:
185:
182:
168:
153:
149:
143:
140:
139:
136:
119:
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