Telescoping - Example

Compute $\sum_{i=1}^{n} \left(\frac{1}{i} - \frac{1}{i+1}\right)$.

Expand: $\left(\frac{1}{1} - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \left(\frac{1}{3} - \frac{1}{4}\right) + ... + \left(\frac{1}{n} - \frac{1}{n+1}\right)$.

The $-\frac{1}{2}$ and $+\frac{1}{2}$ cancel, the $-\frac{1}{3}$ and $+\frac{1}{3}$ cancel, and so on. You're left with $\frac{1}{1} - \frac{1}{n+1} = 1 - \frac{1}{n+1}$.