<s>[INST] Find the sum of the series $\sum_{k=1}^{\infty} \frac {1}{(k)(k+2)(k+4)}$. Problem:
Find the sum of the series $\sum_{k=1}^{\infty} \frac {1}{(k)(k+2)(k+4)}$.
Thoughts
I first tried to write out the series to detect some kind of a pattern, I suspect I need a way of rewriting the expression $\frac {1}{(k)(k+2)(k+4)}$ , but not sure how to proceed. To help visualize it looks like:
$\frac {1} {(1)(3)(5)} + \frac {1} {(2)(4)(6)} +  \frac {1} {(3)(5)(7)} + ...$
 [/INST] $$\frac{1}{k(k+2)(k+4)} = \frac{1}{8} \left( \left( \frac{1}{k} - \frac{1}{k+2} \right) - \left( \frac{1}{k+2} - \frac{1}{k+4} \right) \right).$$
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