Telescope Sum

math

Example

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