Question: Euler-Maclaurin Summation.

Greetings to all.

A basic Euler-MacLaurin Summation recently appeared at this Math.Stackexchange Link . Several users computed an asymptotic expansion for the sum given by sum(1/k^sigma, k=1..n) with sigma a parameter between zero and one. E.g. for sigma=1/3 we find

       2/3
    3 n                     1           1           7            13            247
    ------ + Zeta(1/3) + -------- - --------- + ---------- - ----------- + ------------
      2                     (1/3)       (4/3)         10/3          16/3           22/3
                         2 n        36 n        4860 n       26244 n       590490 n

Comparing this with Euler-Maclaurin Maple produces the following:

> eulermac(1/n^(1/3), n, 8);
       2/3
    3 n         1           1           7            13            247             1
    ------ - -------- - --------- + ---------- - ----------- + ------------ + O(-------)
      2         (1/3)       (4/3)         10/3          16/3           22/3      (28/3)
             2 n        36 n        4860 n       26244 n       590490 n         n

The sign on the second term is incorrect. I would be curious to know what is happening here. BTW Mathematica also yields a plus sign on this term, same as what was obtained at the link.

Thank you for any comments you may have.

Marko Riedel