Compute the sum of the series $\sum_{n=1}^{\infty} \frac{1}{n \cdot 2^n}$

Question

What would be the sum of the following series?

$$\sum_{n=1}^{\infty} \frac{1}{n \cdot 2^n}$$

Thanks

Answer 1

$$\ln(1-x)=-\sum_{n=1}^\infty\frac{x^n}n$$

Can you recognize $x$ here?

See also: Taylor series for $\log(1+x)$ and its convergence

Answer 2

$a_n = \dfrac{(2^{-1})^n}{n} = \dfrac{a^n}{n}$.

Hint: find $\left(\displaystyle \sum_{n=1}^\infty \dfrac{x^n}{n}\right)'$