How to evaluate $\sum_{n=1}^{\infty}\frac{1}{n}(\frac{1}{2^n}-\frac{1}{4^n})$

Question

I want to calculate the infinite summation below : $$\sum_{n=1}^{\infty}\frac{1}{n}(\frac{1}{2^n}-\frac{1}{4^n})$$

but I totally failed to approach.

Thanks for any help.

Wolframalpha gave me the results : $\ln(3/2)$.

Answer 1

Hint. One may recall that $$ -\ln(1-x)=\sum_{n=1}^\infty \frac{x^n}n,\qquad |x|<1, $$ which may be obtained by integrating termwise $$ \frac{1}{1-x}=\sum_{n=0}^\infty x^n,\qquad |x|<1. $$