Can't find the result of $ \sum_{n=0}^{\infty} \frac 1 {n(n+1)(n+2)}$

Question

I can't find the result of:

$$ \sum_{n=0}^{\infty} \frac 1 {n(n+1)(n+2)}$$

From the p-criterion the series converges,but what do we do to find the result?

Answer 1

Hint : $$ \sum_{n=0}^{\infty} \frac {1}{n(n+1)(n+2)}$$ $$\begin {align} = \frac {1}{2}. \sum_{n=0}^{\infty} \left(\frac {1}{n(n+1)}- \frac {1}{(n+2)(n+1)}\right) =\frac {1}{2} \end {align}$$