Series expansion of $\ln(1+\exp(ix))$

Question

I am looking for a way to prove the following identity

$$\ln(1+\exp(ix))= \sum_{n=1}^{\infty}\frac{(-1)^{n+1} \exp(inx)}{n} $$

What I know about the function is the given function is periodic with period $2 \pi$. Can you give a hint on how to get this identity? Thanks in Advance.

Answer 1

This is just the Taylor's series of $\ln 1+z$ $$\ln(1+z)= \sum_{n=1}^{\infty}\frac{(-1)^{n+1} z^n}{n} $$and substituting back $z=\exp(ix)$.