TITLE: $\sum_{n=1}^{\infty}\frac{1}{n^22^n}$ by integration or differentiation
QUESTION [12 upvotes]: There is an infinite sum given:
$$\sum_{n=1}^{\infty}\frac{1}{n^22^n}$$
It should be solved using integration, derivation or both.
I think using power series can help but I don't know how to finish the calculation. Any help will be appreciated!

REPLY [2 votes]: My answer to a duplicate question says:
$$\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}$$
 Integrating from 0 to t we get
$$\int_{0}^{t}\frac{1}{(1-x)}dx=\sum_{n=0}^{\infty}\int_{0}^{t} x^{n}dx$$$$-\ln(1-t)=\sum_{n=1}^{\infty} \frac{t^{n}}{n}$$
Dividing by t and integrating
$$\int_{0}^{0.5}-\frac{\ln(1-t)}{t}dt=\sum_{n=1}^{\infty}\int_{0}^{0.5} \frac{t^{n-1}}{n}dt=\sum_{n=1}^{\infty} \frac{1}{n^{2}2^{n}}$$
This on calculating is $$ \dfrac{\pi^2}{12}-\dfrac{ln^2(2)}{2},$$