I have been reading about Riemann Zeta function and after some manipulation the book states the expression $$ \zeta(s) = (1-2^{1-s})^{-1} \sum\limits_{n=1}^{\infty} (-1)^n n^{-s} $$ which is used to extend the function on $Re(s) >0$.
I am stuck with the expression $\sum\limits_{n=1}^{\infty} (-1)^n n^{-s}$. How to prove that this summation exists for all $Re(s)>0$? The book simply states "alternating series" exist, but I am not sure how to proceed when we assume that $s$ is complex. Any help would be appreciated.