<s>[INST] Find the sum of the series $\sum \frac{1}{n(n+1)(n+2)}$ I got this question in my maths paper

Test the condition for convergence of $$\sum_{n=1}^\infty \frac{1}{n(n+1)(n+2)}$$
  and find the sum if it exists.

I managed to show that the series converges but I was unable to find the sum. Any help/hint will go a long way. Thank you.
 [/INST] Writing 
$$\frac{1}{n(n+1)(n+2)} = \frac{1}{n+1}\left(\frac{1}{2n} - \frac{1}{2(n+2)}\right) = \frac{1}{2n(n+1)} - \frac{1}{2(n+1)(n+2)},$$
we see that the sum telescopes to 
$$\frac{1}{2(1)(2)} = \frac{1}{4}.$$
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