How could I prove that as $ \epsilon \to 0 $ the regularized series goes as
$$ \sum _{n=1}^{\infty} \frac{\exp(-n\epsilon)}{n}=-\log(1-e^{-\epsilon}) $$
and how could I prove that the finite part
$$ \text{F.P}\sum _{n=1}^{\infty} \frac{\exp(-n\epsilon)}{n}= \gamma $$
Euler-Mascheroni constant.
how about the series $ \text{F.P} \sum _{n=1}^{\infty} n^{s}\exp(-n\epsilon)= \zeta (-s) $