Given $\zeta(s) = 1^{-s} + 2^{-s} + 3^{-s}...$
$2^{-s}\zeta(s) = 2^{-s} + 4^{-s} + 6^{-s}...$
$\zeta(s)(1-2^{1-s})=1^{-s} - 2^{-s} + 3^{-s} - 4^{-s}...$ which converges for $0 < \Re(s) < 1$.
Is it therefore valid to calculate the output of this convergent series and use it to find values of $\zeta(s)$ in the range $0 < \Re(s) < 1$? If so then is this valid for $\zeta(0)$ also?
Thanks for any help! It will be much appreciated!