Telescope sum

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Pushing together a telescope - namesake of the telescope sum
A collapsible telescope

A telescoping series is in the mathematics a finite sum of differences (except the first and the last) cancel each other, wherein each two neighboring members. This process is called telescoping a sum . The term is derived from pushing two or more cylindrical tubes into one another .

If there is a consequence , then is

a telescope sum. If you can write a sum as a telescope sum, its evaluation is simplified:


A series whose partial sums are telescope sums is called a telescope series . A row of telescopes

is convergent if and only if converges to a limit value . The sum of the series is then the same .

The same applies to a product:

is a telescope product , so to speak .

The situation is more complicated when the telescope runs over three (or more) consecutive links (see example).

In number theory telescopic sums represent an important tool.


The partial fraction decomposition of can be obtained using, for example
It follows
  • three times the telescope sum:
Alternatively, this follows for by differentiation from the first example with the help of the quotient rule :
This is an important application example of differential calculus as a calculus for term transformation .


  • Rolf Walter: Introduction to Analysis. Volume 1. Walter de Gruyter, Berlin et al. 2007, ISBN 978-3-11-019539-2 , pp. 31–32 ( excerpt in the Google book search).
  • Harro Heuser : Textbook of Analysis. Volume 1. 6th edition, unchanged reprint of the 5th revised edition. Teubner, Stuttgart et al. 1989, ISBN 3-519-42221-2 , pp. 91, 94, 194.

Web links

  • Telescoping Sum on PlanetMath
  • Po-Shen Loh: Telescoping Sums and Products (PDF; 66 kB) - examples of telescope sums and telescope products
  • Philippe B. Laval: Telescoping Sums (PDF; 95.1 kB) - Derivation of a general theorem for telescoping sums of the form