Find the sum of the
$$\sum_{n=1}^\infty \frac{1}{n2^n}$$
Edit after YiFan answer
If we take $a_n=1$ , we get:
$$f(x)=\int\frac{1}{x} \cdot \frac{x}{1-x}$$ $$f(x)=\ln\frac{1}{|x-1|}$$ $$f(\frac{1}{2})= \ln{2}$$
what is equal to wolfram answer
Thank you for the help :).
Hint. Let $$f(x)=\sum_{n=1}^\infty\frac{x^n}{n},$$ then we want to find $f(1/2)$. Make the following two obervations: first that $$\int\frac1x\sum a_nx^n\,dx=\sum\int a_nx^{n-1}\,dx=\sum\frac{a_n}{n}x^{n}$$ so long as exchanging $\int$ and $\sum$ is justified (ask yourself: when is this true?) and then that $$\sum_{n=1}^\infty x^n=\frac{x}{1-x}.$$