<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] An alternative approach:
$$\begin{eqnarray*} \sum_{k\geq 1}\frac{1}{k(k+2)(k+4)}&=&\frac{1}{8}\sum_{k\geq 1}\left(\frac{1}{k}-\frac{2}{k+2}+\frac{1}{k+4}\right)\\&=&\frac{1}{8}\sum_{k\geq 1}\int_{0}^{1}(1-2x^2+x^4)x^{k-1}\,dx\\&=&\frac{1}{8}\int_{0}^{1}\frac{(1-x^2)^2}{1-x}\,dx\\&=&\frac{1}{8}\int_{0}^{1}(1+x-x^2-x^3)\,dx\\&=&\frac{1}{8}\left(\frac{1}{1}+\frac{1}{2}-\frac{1}{3}-\frac{1}{4}\right)=\color{red}{\frac{11}{96}}.\end{eqnarray*}$$
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