0

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.

Kr Dpk
  • 483
  • Hint: https://en.m.wikipedia.org/wiki/Summation_by_parts – Aphelli Dec 14 '21 at 15:32
  • Alternating series holds only for $s$ real. For $s$ complex note that $$\sum_{n\ge 1} (2n-1)^{-s}-(2n)^{-s}=\sum_{n\ge 1} \int_{2n-1}^{2n}sx^{-s-1}dx$$ is absolutely convergent for $\Re(s) >0$ – reuns Dec 14 '21 at 19:37

0 Answers0