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.

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