Telescoping Sum

analysis

Definition

Telescope Sum

A telescoping sum is a finite sum in which pairs of consecutive terms partly cancel each other, leaving only parts of the initial and final terms.

Examples

Example: $\frac{n}{(n+1)!}$

Given a sequence $a_n=\frac{n}{(n+1)!}$ and its series:

$$\begin{array}{rl}\sum _{k=0}^n{a}_k& =\sum _{k=0}^n\frac{k}{(k+1)!}\\ & =\sum _{k=0}^n\frac{k+1-1}{(k+1)!}\\ & =\sum _{k=0}^n\left(\frac{k+1}{(k+1)!}-\frac{1}{(k+1)!}\right)\\ & =\sum _{k=0}^n\left(\frac{1}{k!}-\frac{1}{(k+1)!}\right)\\ & =\left(\frac{1}{1!}-\frac{1}{2!}\right)+\left(\frac{1}{2!}-\frac{1}{3!}\right)+\dots +\left(\frac{1}{(k-1)!}-\frac{1}{k!}\right)+\left(\frac{1}{k!}-\frac{1}{(k+1)!}\right)\\ & =\frac{1}{1!}-\frac{1}{(n+1)!}\end{array}$$