<s>[INST] sum of infinite series with telescopes I need help finding the sum of the infinite series
$$\sum_{k=1}^\infty \frac{1}{n(n+1)(n+2)}$$
I have used the partial fraction decomposition to get this as the sum of 
$$\frac{-1}{k+1}+\frac{1}{2(k+2)}+\frac{1}{2k}$$
but don't know where to go from here.
Thanks!
 [/INST] Hint. First observe that $$
\frac{1}{n(n+1)(n+2)}=\frac{1}{2}\left(\frac{1}{n}-\frac{1}{n+1}\right)-\frac{1}{2}\left(\frac{1}{n+1}-\frac{1}{n+2}\right)
$$ and conclude by telescoping terms.
You end up with
$$
\sum_{n=1}^{N}\frac{1}{n(n+1)(n+2)}=\frac{1}{2}\left(1-\frac{1}{N+1}\right)-\frac{1}{2}\left(\frac{1}{2}-\frac{1}{N+2}\right)
$$ and then let tend $N$ to $+\infty$.
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