Problem statement:
Let $f(z) :$
$$f(z) = \sum_{n=1}^{\infty}{(-1)^{n-1}\cdot\left(\frac{(z-1)^{n}}{n}\right)} $$
Q: How can we prove that the previous mapping is a branch of logarithm on $\mathbb{D}(1,1)\;$ (open disk)
Thanks in advance for your help.
EDIT: If $U \subset \mathbb{C}^{*}$ an open set and $f$ a branch of logarithm prove that :
i) $ f$ is analytic
ii) $f'(z) = \frac{1}{z} $