<s>[INST] Difficult Telescoping Series Finding the explicit sum of a telescoping series with two factors in the denominator is not a problem: we split the fractions in the difference of two pieces.
But what about 2+ factors
\begin{equation}
\lim_{n\to\infty} \sum \frac{1}{k(k+2)(k+4)}
\end{equation}
 [/INST] hint: we can do telescoping inside teloscoping as follows:
$\dfrac{1}{n(n+2)(n+4)}= \dfrac{1}{4}\left(\dfrac{1}{n(n+2)} - \dfrac{1}{(n+2)(n+4)}\right)$, and split again: $\dfrac{1}{n(n+2)} = \dfrac{1}{2}\left(\left(\dfrac{1}{n} - \dfrac{1}{n+1}\right) + \left(\dfrac{1}{n+1} - \dfrac{1}{n+2}\right)\right)$. Do the same for the other term and there are a total of $4$ "mini" telescoping sums to be evaluated.
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