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.

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

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

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

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

Title	telescoping sum
Canonical name	TelescopingSum
Date of creation	2013-03-22 14:25:18
Last modified on	2013-03-22 14:25:18
Owner	cvalente (11260)
Last modified by	cvalente (11260)
Numerical id	8
Author	cvalente (11260)
Entry type	Definition
Classification	msc 40A05
Defines	telescope