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As long as the starting point for the index is finite and the domain of the index is countable (and – for simplivity – non-empty), should not all partial sums have a finite number of terms after cancellation? Moreover, after commutation, regrouping, and some number of evaluations, the number of terms
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From the pitfall section: "is not correct because this regrouping of terms is invalid unless the individual terms converge to 0;" The alternating harmonic series has individual terms which converge to zero, but regrouping of terms is not permissable there. Shouldn't this say that the partial sum of
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While I agree that the title is misleading, a quick websearch notes that many math educational sites like Khan
Academy and Brilliant refer to them as Telescoping Series. However, the esteemed Wolfram Alpha refers to them as Telescoping Sums. For this reason, I will add the terminology to the page.
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It is easy to create an example where the terms and the grouped terms tend to zero which does not converge. Create a series as follows: 1 - 1 + 1/2 + 1/2 - 1/2 - 1/2, + 1/4 + 1/4 + 1/4 + 1/4 - 1/4 - 1/4 - 1/4 + ... where there are n terms 1/2^n followed by n terms -1/2^n. It is easy to group this
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I have changed the opening paragraph. The salient feature of a telescoping series is that the general term is presented in a way such that consecutive terms cancel---again, it is about the *presentation* of the series, not the series itself. Any series *can* be rewritten as a telescoping sum.
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I would take "telescoping" to mean that in finite partial sums every term cancels except the first and the last, or the last two, or some bounded number of terms. ("Bounded" means never exceeding some number that does not grow as the number of terms being summed grows.) The article isn't very
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and others. The questions for this article are, what distinguishes "telescoping", when and where did the term originate, and how is it used today. With respect to current use, Gosper and others have made significant contributions to algorithms used in modern symbolic
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The restriction that terms have to be alternately positive and negative does avoid the problem I mentioned. Alternatively, one can have an arbitrary series and group terms together by sign (resulting in a collapsed series whose terms alternate in sign).
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But your counterexample is not a telescoping series, as the terms don't go positive, negative, positive, negative... If you arrange them that way, you have 1 - 1 + 1/2 - 1/2 + 1/2 - 1/2 + 1/4 - 1/4 + ..., which is truly telescoping and certainly
1355:{\displaystyle \prod _{i=m}^{n}x_{i}={\frac {y_{m}}{y_{m-1}}}\cdot {\frac {y_{m+1}}{y_{m}}}\cdot {\frac {y_{m+2}}{y_{m+1}}}\cdot \,\,\cdots \,\,\cdot {\frac {y_{n-1}}{y_{n-2}}}\cdot {\frac {y_{n}}{y_{n-1}}}={\frac {y_{n}}{y_{m-1}}}.}
1011:{\displaystyle \prod _{i=m}^{n}x_{i}={\frac {y_{m}}{y_{m+1}}}\cdot {\frac {y_{m+1}}{y_{m+2}}}\cdot {\frac {y_{m+2}}{y_{m+3}}}\cdot \,\,\cdots \,\,\cdot {\frac {y_{n-1}}{y_{n}}}\cdot {\frac {y_{n}}{y_{n+1}}}={\frac {y_{m}}{y_{n+1}}}.}
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Is there a possibility of a proof that relates for telescoping series? By that I mean, is there a formula for finding out what the final value is when it converges? If anyone really wants to talk about it, you can discuss it
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A criterion that is adequate for grouping to be valid is that the absolute sum of the terms in each group tends to zero over the sequence of groups. This criterion is true for the valid example given.
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that the term has been found in a 1957 document on mathscinet. Someone is telling me that it's in a math dictionary published in 1949. More to follow when I get more information. My guess is the
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The pitfalls section is bad as it stands. The argument used for the valid sum could be used for the invalid example. It is necessary to use a convergence criterion that distinguises between them.
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And to be clear, I completely agree that series were being rearranged long before Gosper. One of the challenges of formalization was to understand what rearrangements were safe, as pursued by
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converge to 0. So there is nothing wrong with the convergence criterion stated in the article, except that it only works for those series covered by the article. --
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What is more, a partial sum, by definition, only consists of a finite number of terms. So, I'm completely lost reading the first sentence of the article.
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Oops, sometimes, it cannot be reduced to more than 2 terms. And, I suppose, it could be reduced to 0 terms, if all of them cancel somehow (easy example:
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introduced the idea of telescoping. He is one of the pioneers of computer symbolic mathematics programs, having contributed to both
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I think it would be to include some practical applications of telescoping series in this article, but so far I have not found any.
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can be reduced from the original N to any positive integer number of terms. I am not sure that this definition is really correct.
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I'm removing it as it doesn't make any sense whatsoever. Someone's inability to do maths is not a pitfall of telescoping series.
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What!!!??? How could he have introduced this idea if his life was so recent that he worked with electronic computers??
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I can't find any reliable reference to the term "telescoping product". It would be one of the following:
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I propose putting the second occurance first, then putting something like, "This is useful to prove
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for further details. I apologize if my previous brief remark confused or annoyed the historians! --
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Knowledge. If you would like to participate, please visit the project page, where you can join
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I would aim more for "partial sums have a pattern of cancellation between terms" or something.
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The extraordinary generality of hypergeometric series and the power of these modern algorithms
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with each grouped term being zero, but the original series oscillates rather than converging.
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Maybe there are indeed divergent examples of telescoping series. But this one is just wrong.
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originated between 1850 and 1910 -- but that's just a guess. Telescoping sums were used by
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I've added an application. I think others can be found by clicking on "what links here".
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Not all telescoping sums are infinite series; some are finite. Should this be moved to
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So far there is NOTHING wrong with this statement: 0 = ∑0 = ∑(1 -1) = ∑(-1 +1)
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What the final value is, is certainly stated explicitly in the article.
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precise about that. I'll think about how best to phrase a definition.
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349:"Example 1" is essentially the same as the earlier example.
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but they will probably tell you to do it yourself, using
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allows the discovery of new and valuable identities. See
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the sum tends to 1." Then removing that occurance.
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662:contribs
650:unsigned
628:unsigned
532:Pitfalls
510:contribs
502:Panchaos
498:unsigned
289:Abramov
407:Rewrite
209:Macsyma
139:on the
1470:Madyno
1455:IbexNu
1399:IbexNu
599:Elroch
547:Elroch
467:x42bn6
461:limits
436:Proof?
422:x42bn6
392:x42bn6
218:cites
199:Gosper
36:scale.
447:EdBoy
336:KSmrq
291:et al
256:Euler
224:KSmrq
1496:talk
1474:talk
1459:talk
1403:talk
1371:talk
658:talk
636:talk
603:talk
587:talk
571:talk
563:does
551:talk
506:talk
471:Talk
443:here
427:Talk
397:Talk
252:term
211:and
187:talk
1453:).
1365:--
1021:or
567:Jao
463:.
445:. —
419:.
322:A=B
299:PDF
170:.)
131:Low
1511::
1498:)
1476:)
1461:)
1435:−
1420:∑
1405:)
1373:)
1340:−
1307:−
1284:⋅
1274:−
1258:−
1245:⋅
1238:⋯
1231:⋅
1192:⋅
1159:⋅
1149:−
1096:∏
1065:−
940:⋅
920:−
907:⋅
900:⋯
893:⋅
854:⋅
815:⋅
752:∏
664:)
660:•
638:)
605:)
589:)
573:)
553:)
512:)
508:•
377:∞
374:→
189:)
1494:(
1472:(
1457:(
1441:)
1438:n
1432:n
1429:(
1424:n
1401:(
1369:(
1350:.
1343:1
1337:m
1333:y
1327:n
1323:y
1317:=
1310:1
1304:n
1300:y
1294:n
1290:y
1277:2
1271:n
1267:y
1261:1
1255:n
1251:y
1224:1
1221:+
1218:m
1214:y
1208:2
1205:+
1202:m
1198:y
1185:m
1181:y
1175:1
1172:+
1169:m
1165:y
1152:1
1146:m
1142:y
1136:m
1132:y
1126:=
1121:i
1117:x
1111:n
1106:m
1103:=
1100:i
1068:1
1062:i
1058:y
1052:i
1048:y
1042:=
1037:i
1033:x
1006:.
999:1
996:+
993:n
989:y
983:m
979:y
973:=
966:1
963:+
960:n
956:y
950:n
946:y
933:n
929:y
923:1
917:n
913:y
886:3
883:+
880:m
876:y
870:2
867:+
864:m
860:y
847:2
844:+
841:m
837:y
831:1
828:+
825:m
821:y
808:1
805:+
802:m
798:y
792:m
788:y
782:=
777:i
773:x
767:n
762:m
759:=
756:i
724:1
721:+
718:i
714:y
708:i
704:y
698:=
693:i
689:x
656:(
634:(
601:(
585:(
569:(
549:(
516:.
504:(
371:N
327:)
301:)
185:(
143:.
42::
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