Greetings to all.

I would like to share a brief observation concerning my experiences with the Euler-Maclaurin summation routine in **Maple 17 (X86 64 LINUX)**. The following Math StackExchange Link shows how to compute a certain Euler-MacLaurin type asymptotic expansion using highly unorthodox divergent series summation techniques. The result that was obtained matches the output from **eulermac** which is definitely good to know. What follows is the output from said routine.

> eulermac(1/(1+k/n),k=0..n,18);
1 929569 3202291 691 O(1)
O(- ---) - ----------- + ----------- - --------- + 1/1048576 ----
19 15 17 11 19
n 2097152 n 1048576 n 32768 n n
n
/
174611 5461 31 | 1 17 1
- -------- + --------- + ------- + | ------- dk - ------- + ------
19 13 9 | 1 + k/n 7 5
6600 n 65536 n 4096 n / 4096 n 256 n
0
1 1
- ------ + ---- + 3/4
3 16 n
128 n

While I realize that this is good enough for most purposes I have two minor issues.

- One could certainly evaluate the integral without leaving it to the user to force evaluation with the
**AllSolutions** option. One can and should make use of what is known about n and k. In particular one can check whether there are singularities on the integration path because we know the range of *k/n*.
- Why are there two order terms for the order of the remainder term? There should be at most one and a coefficient times an
*O(1)* term makes little sense as the coefficient would be absorbed.

You might want to fix these so that the output looks a bit more professional which does enter into play when potential future users decide on what CAS to commit to. Other than that it is a very useful routine even for certain harmonic sum computations where one can use Euler-Maclaurin to verify results.

Best regards,

Marko Riedel