The series $\sum_{n=1}^{\infty}\frac{1}{n+1}$ diverge. What about $\sum_{n=1}^{\infty}\frac{1}{(n+1) \times (n+2)}$?

Question

The series $\sum_{n=1}^{\infty}\frac{1}{n+1}$ diverge. What about $\sum_{n=1}^{\infty}\frac{1}{(n+1) \times (n+2)}$?

Answer 1

Hint: $$\frac1{(n+1)(n+2)} = \frac1{n+1} - \frac1{n+2}.$$

Answer 2

It is just $$=\left(\frac12-\frac13\right)+\left(\frac13-\frac14\right)+...=\frac12.$$