PlanetMath: telescoping sum

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telescoping sum

(Definition)

A telescoping sum is a sum in which cancellation occurs between subsequent terms, allowing the sum to be expressed using only the initial and final terms.

Formally a telescoping sum is or can be rewritten in the form

$\displaystyle S= \sum_{n=\alpha}^{\beta}\left(a_n - a_{n+1}\right) = a_\alpha - a_{\beta+1}$

where is a sequence.

Example:

Define $S(N) = \sum_{n=1}^{N} \frac{1}{n(n+1)}$ . Note that by partial fractions of expressions:

$\displaystyle \frac{1}{n(n+1)}= \frac{1}{n} - \frac{1}{n+1}$

and thus $a_n = \frac{1}{n}$ in this example.

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

$\displaystyle = \left( 1 - \frac{1}{2} \right) + \cdots + \left( \frac{1}{n} - ... ... - \frac{1}{n+2} \right) + \cdots + \left( \frac{1}{N} - \frac{1}{N+1} \right)$

$\displaystyle = 1 + \left( - \frac{1}{2} + \frac{1}{2} \right) + \cdots + \left( -\frac{1}{n+1} +\frac{1}{n+1} \right) + \cdots - \frac{1}{N+1}$

$\displaystyle = 1 - \frac{1}{N+1}$

"telescoping sum" is owned by cvalente. [ full author list (2) | owner history (1) ]

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Also defines:

telescope

Attachments:: example of telescoping sum (Example) by pahio

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Cross-references: partial fractions of expressions, sequence, terms, sum
There are 6 references to this entry.

This is version 5 of telescoping sum, born on 2004-06-17, modified 2008-05-03.
Object id is 5929, canonical name is TelescopingSum.
Accessed 4982 times total.

Classification:

AMS MSC:

40A05 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences)

Pending Errata and Addenda

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