<s>[INST] Summation of Infinite Series $\sum_{n=1}^{\infty} \frac{1}{n 2^{2n+1}}$ Show That :
$$\sum_{n=1}^{\infty} \frac{1}{n 2^{2n+1}} = \ln \left(\frac{2}{\sqrt{3}}\right)$$
I could show convergence. (I dont need to show that this converges). However I couldn't figure how to show the value.
 [/INST] $$
\begin{align*}
\sum_{n=1}^{\infty} \frac{y^n}{n} &= -\ln \left(1-y\right) \hspace{15pt} {\textit{apply }} \hspace{5pt} y=\frac{1}{x^2}\\
\sum_{n=1}^{\infty} \frac{1}{n x^{2n}} &= -\ln \left(1-\frac{1}{x^2}\right) \\
\sum_{n=1}^{\infty} \frac{1}{n 2^{2n+1}} &= \frac{1}{2}\sum_{n=1}^{\infty} \frac{1}{n 2^{2n}} = -\frac{1}{2}\ln \left(1-\frac{1}{4}\right)\\
\end{align*}
$$
Do the rest to simplify and get what you want.
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