How to calculate this sum $$\sum_{n=1}^{\infty} \frac{H_n \cdot H_{n+1}}{(n+1)(n+2)}$$
Attempt
The series telescopes. We have $$=\frac{H_n \cdot H_{n+1}}{(n+1)(n+2)} = \frac{H_n \cdot H_{n+1}}{n+1} - \frac{H_n \cdot H_{n+1}}{n+2}$$
$$=\frac{H_n(H_n + \frac{1}{n+1})}{n+1} - \frac{(H_{n+1} - \frac{1}{n+1}) \cdot H_{n+1}}{n+2}$$
$$=\frac{H_n^2}{n + 1} - \frac{H_{n+1}^2}{n + 2} + \frac{H_n}{(n + 1)^2}+ \frac{H_{n+1}}{(n + 1)(n + 2)}$$
$$=\frac{H_n^2}{n + 1} - \frac{H_{n+1}^2}{n + 2} + \frac{H_{n+1} - \frac{1}{n+1}}{(n+1)^2} + - \frac{H_{n+1}}{n+1} - \frac{H_{n+1}}{n+2}$$
$$=\frac{H_n^2}{n + 1} - \frac{H_{n+1}^2}{n + 2} + \frac{H_{n+1}}{(n+1)^2} - \frac{1}{(n+1)^3} + \frac{H_{n + 1}}{n + 1} - \frac{H_{n + 2}}{n + 2} + \frac{1}{{(n + 2)^2}}$$
It follows that $$\sum_{n=1}^{\infty} \frac{H_n \cdot H_{n+1}}{(n+1)(n+2)} = \sum_{n=1}^{\infty} \left(\frac{H_{n}^2}{n + 1} - \frac{H_{n+1}^2}{(n+2)^2}\right) + \sum_{n=1}^{\infty} \frac{H_{n+1}}{(n+1)^2} - \sum_{n=1}^{\infty} \frac{1}{(n+1)^3}$$