Find the sum of series $\sum_{n=2}^\infty\frac{1}{n(n+1)^2(n+2)}$ How to find the sum of series $\sum_{n=2}^\infty\frac{1}{n(n+1)^2(n+2)}$ in the formal way? Numerically its value is $\approx 0.0217326$ and the partial sum formula contains the first derivative of the gamma function (by WolframAlpha).

Partial fractions:  $$\frac{1}{n(n+1)^2(n+2)} = \frac{1}{2} \left( \frac{1}{n} - \frac{1}{n+2} \right) - \frac{1}{(n+1)^2},$$ the first part being telescoping, and the last part being related to $\zeta(2)$.
