Q:

Evaluating the limit of a series

I am trying to evaluate the limit of the following series:
$$\sum_{k=1}^\infty \frac{k}{(k+1)(k+2)(k+3)}$$
I have already decomposed the fraction into the following:
$$\frac{2}{k+2}-\frac{3}{2(k+3)}-\frac{1}{2(k+1)}$$
I am stuck at this point. I don't see what positive terms cancel each other out with their negative counterparts between the three. Any help would be appreciated. Thanks in advance.

A:

Your partial fraction decomposition is incorrect, and has an extra minus sign in front of the $3$.  You just have to write out the terms, and you are left with $$\frac 12\left(-\frac 12+\frac 43-\frac 13\right)$$ with everything else cancelling, so the answer is $$\frac 14$$

