# telescoping sum

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

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

where $a_{n}$ is a sequence.

Example:

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

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

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

 $S(N)=\sum_{n=1}^{N}\left(\frac{1}{n}-\frac{1}{n+1}\right)$
 $=\left(1-\frac{1}{2}\right)+\cdots+\left(\frac{1}{n}-\frac{1}{n+1}\right)+% \left(\frac{1}{n+1}-\frac{1}{n+2}\right)+\cdots+\left(\frac{1}{N}-\frac{1}{N+1% }\right)$
 $=1+\left(-\frac{1}{2}+\frac{1}{2}\right)+\cdots+\left(-\frac{1}{n+1}+\frac{1}{% n+1}\right)+\cdots-\frac{1}{N+1}$
 $=1-\frac{1}{N+1}$
Title telescoping sum TelescopingSum 2013-03-22 14:25:18 2013-03-22 14:25:18 cvalente (11260) cvalente (11260) 8 cvalente (11260) Definition msc 40A05 telescope