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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.

Gottfried Helms
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Jose Garcia
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  • It would be better if the calligraphic [R]-symbol were at the sum in the first formula (like it is in the article on the last page) to indicate, that the sum is not meant in the standard way (in which case your first formula were false) but it is meant in Ramanujan's special way – Gottfried Helms Feb 06 '14 at 14:49
  • -1 because still without correct modifyer in the first formula. The reference to the $zeta$-function looks as if in the Delabaere-paper were an error. In that paper is not the Euler/Riemann-zeta-function but an explicite made modification at that plcase and this is indicated there as such. – Gottfried Helms Feb 12 '14 at 16:56

1 Answers1

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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.)

mrf
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