<s>[INST] Evaluate $\sum_ \limits{n=1}^{\infty} \frac{n}{1 \cdot 3 \cdot 5 \cdots (2n+1) } $ $$\sum_ \limits{n=1}^{\infty} \frac{n}{1 \cdot 3 \cdot 5 \cdots (2n+1) } $$
$$1 \cdot 3 \cdot 5 \cdots (2n+1) = \frac{1 \cdot 2 \cdot 3 \cdots (2n+2)}{2 \cdot 4 \cdot 6 \cdots (2n+2)} = \frac{(2n+2)!}{2^{n+1} \cdot (n+1)!} $$
But the follow up gives me $\infty$. How to approach this type of exercises?
The result should be $1/2$
 [/INST] HINT:
Note that
$$\begin{align}
\frac{n}{(2n+1)!!}&=\frac12\left( \frac{2n+1-1}{(2n+1)!!}\right)\\\\&=\frac12\left(\frac{1}{(2n-1)!!}-\frac1{(2n+1)!!}\right)
\end{align}$$
Now, telescope.
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