Real series/Sum 1 over k(k+1)/Example

We want to compute the series

$\sum _{k=1}^{\infty }{\frac {1}{k(k+1)}}.$

For this, we give a formula for the ${}n$-th partial sum. We have

${}s_{n}=\sum _{k=1}^{n}{\frac {1}{k(k+1)}}=\sum _{k=1}^{n}{\left({\frac {1}{k}}-{\frac {1}{k+1}}\right)}=1-{\frac {1}{n+1}}={\frac {n}{n+1}}\,.$

This sequence converges to ${}1$, so that the series converges and its sum equals ${}1$.