I have been studying the Riemann zeta function and I came across such a representation of zeta function
$\zeta(s)=(1-2^{1-s})^{-1} \sum_{n=1}^{\infty} (-1)^{n-1} n^{-s}$.
Then it says in the book that this alternating series converges for $Re(s) > 0$. And there is my question. How can we show that this series converges in that region? We can't use the alternating series test because we deal with complex numbers, but please correct me if I'm wrong, so how can we do it? Also, could anyone explain to me or recommend any book to read why the imaginary part does not have any impact on convergence of a series? Thanks!
