In mathematics, telescoping series is an informal expression referring to a series whose sum can be found by exploiting the circumstance that nearly every term cancels with a succeeding or preceding term. Such a technique is also known as the method of differences.

For example, the series

$\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{1\}\{n(n+1)\}$

simplifies as

$$

\sum_{n=1}^\infty \frac{1}{n(n+1)} = \sum_{n=1}^\infty \left( \frac{1}{n} - \frac{1}{n+1} \right)\,

$=\; \backslash left(1\; -\; \backslash frac\{1\}\{2\}\backslash right)$

+ \left(\frac{1}{2} - \frac{1}{3}\right) + \cdots\,

$=\; 1\; +\; \backslash left(-\; \backslash frac\{1\}\{2\}\; +\; \backslash frac\{1\}\{2\}\backslash right)$

+ \left( - \frac{1}{3} + \frac{1}{3}\right) + \cdots = 1. \,

A pitfall

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While telescoping is a neat technique, there are pitfalls to watch out for:

$0\; =\; \backslash sum\_\{n=1\}^\backslash infty\; 0\; =\; \backslash sum\_\{n=1\}^\backslash infty\; (1-1)\; =\; 1\; +\; \backslash sum\_\{n=1\}^\backslash infty\; (-1\; +\; 1)\; =\; 1\backslash ,$

is not correct because regrouping of terms is invalid unless the individual terms converge to 0. The way to avoid this error is to find the sum of the first N terms first and then take the limit as N approaches infinity:

$$

\sum_{n=1}^N \frac{1}{n(n+1)} = \sum_{n=1}^N \left( \frac{1}{n} - \frac{1}{n+1} \right) \,

$=\; \backslash left(1\; -\; \backslash frac\{1\}\{2\}\backslash right)$

+ \left(\frac{1}{2} - \frac{1}{3}\right) + \cdots + \left(\frac{1}{N} - \frac{1}{N+1}\right)\,

$=\; 1\; +\; \backslash left(-\; \backslash frac\{1\}\{2\}\; +\; \backslash frac\{1\}\{2\}\backslash right)$

+ \left( - \frac{1}{3} + \frac{1}{3}\right) + \cdots + \left(-\frac{1}{N} + \frac{1}{N}\right) - \frac{1}{N+1} \,

$=\; 1\; -\; \backslash frac\{1\}\{N+1\}\backslash to\; 1\backslash \; \backslash mathrm\{as\}\backslash \; N\backslash to\backslash infty.\backslash ,$

More examples

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• Many trigonometric functions also admit representation as a difference, which allows telescopic cancelling between the consequent terms.

$\backslash sum\_\{n=1\}^N\; \backslash sin\backslash left(n\backslash right)\; =\; \backslash sum\_\{n=1\}^N\; \backslash frac\{1\}\{2\}\; \backslash csc\backslash left(\backslash frac\{1\}\{2\}\backslash right)\; \backslash left(2\backslash sin\backslash left(\backslash frac\{1\}\{2\}\backslash right)\backslash sin\backslash left(n\backslash right)\backslash right)$

$=\backslash frac\{1\}\{2\}\; \backslash csc\backslash left(\backslash frac\{1\}\{2\}\backslash right)\; \backslash sum\_\{n=1\}^N\; \backslash left(\backslash cos\backslash left(\backslash frac\{2n-1\}\{2\}\backslash right)-\backslash cos\backslash left(\backslash frac\{2n+1\}\{2\}\backslash right)\backslash right)$

$=\backslash frac\{1\}\{2\}\; \backslash csc\backslash left(\backslash frac\{1\}\{2\}\backslash right)\; \backslash left(\backslash cos\backslash left(\backslash frac\{1\}\{2\}\backslash right)-\backslash cos\backslash left(\backslash frac\{2N+1\}\{2\}\backslash right)\backslash right).$

• Some sums of the form

$\backslash sum\_\{n=1\}^N\; \{f(n)\; \backslash over\; g(n)\},$

where f and g are polynomial functions whose quotient may be broken up into partial fractions, will fail to admit summation by this method. In particular, we have

$\backslash sum^\backslash infty\_\{n=0\}\backslash frac\{2n+3\}\{(n+1)(n+2)\}$

$=\backslash sum^\backslash infty\_\{n=0\}\backslash left(\backslash frac\{1\}\{n+1\}+\backslash frac\{1\}\{n+2\}\backslash right)$

$=\backslash frac\{1\}\{1\}+\backslash frac\{1\}\{2\}+\backslash frac\{1\}\{2\}+\backslash frac\{1\}\{3\}+\backslash frac\{1\}\{3\}+\backslash frac\{1\}\{4\}+\backslash cdots+\backslash frac\{1\}\{n-1\}+\backslash frac\{1\}\{n\}+\backslash frac\{1\}\{n\}+\backslash frac\{1\}\{n+1\}+\backslash frac\{1\}\{n+1\}+\backslash frac\{1\}\{n+2\}+\backslash cdots$

$=\backslash infty.$

The problem is that the terms do not cancel.

• A special kind of series

Let $k$ be a positive integer, then

$\backslash sum^\backslash infty\_\{n=1\}\; \{\backslash frac\{1\}\{n(n+k)\}\}\; =\; \backslash frac\{H\_k\}\{k\}$

where $H\_k$ is the k-th harmonic number.

Mathematical series

Teleskopsumme | Serie telescopica | 伸縮和