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The following sum always gives -1 if k is undefined but different values when k is defined. Why?

Sum[DifferenceDelta[n^k, n], {n, 1, Infinity}, Regularization -> Dirichlet]
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    $\begingroup$ Do not use the bugs tag until what you've observed has been confirmed by other users. $\endgroup$ Commented Apr 24, 2017 at 13:48

1 Answer 1

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Because:

DifferenceDelta[n^k, n]

gives:

n^k + (1 + n)^k

with k unspecified

Sum[  n^k,{n,1,Infinity}]

is (generically)

-Zeta[-k]

while

Sum[ (1+n)^k,{n,1,Infinity}]

is

-1 + Zeta[-k]

hence

Sum[DifferenceDelta[n^k, n], {n, 1, Infinity}]

gives -1.

The reason for this is that this is the answer for all Complex k for which the sum converges.

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