Determine whether the infinite series $\sum _{n=1}^{\infty }\frac{1}{n^2+n}$ converges or diverges. If it converges, find its sum.

Step-by-step solution

1. Factor the denominator: $n^2+n=n(n+1)$.
2. Rewrite each term as $\frac{1}{n(n+1)}=\frac{1}{n}-\frac{1}{n+1}$ (partial fraction decomposition).
3. The series becomes a telescoping series: ${\sum }_{n=1}^N(\frac{1}{n}-\frac{1}{n+1})=1-\frac{1}{N+1}$.
4. Take the limit as $N\to \infty$: ${\lim }_{N\to \infty }(1-\frac{1}{N+1})=1$.
5. Hence the infinite series converges and its sum is $1$.

Concept tested: Telescoping series convergence

Published 7 October 2026 · How this solution was prepared and checked