according to the paper from delabaere
http://algo.inria.fr/seminars/sem01-02/delabaere2.pdf
$$ \sum_{n=1}^{\infty}n^{k}= \zeta (-k)+ \frac{1}{k+1} $$
and $ \sum_{n=1}^{\infty}n^{-1}= \gamma $
but shouldn't all the results for a certain divergent series agree ? i mean there is an extra term $ \frac{1}{k+1} $ inside the divergent series although the function
$$ \phi (s)= \zeta (-s)+ \frac{1}{k+1} $$
is perfectly well defined and has no poles.
Different summation methods for a divergent series may very well give different results. (Although the most common summations methods do agree on series where they are applicable; for example Abel summation and Cesàro summation give the same results when both are defined, but Abel summation is more general.)