Symbolic summation (was: levin versus ratsum)

Hi Richard,

Thanks for chiming in. The point is that what seems to be missing
in maxima is something like Closedform (-)! As a start we were
discussing  just the easy case of summing rational functions.
I think the current version of maxima does not even simplify
a trivial telescoping sum like sum_k 1/k(k+1).

Yes as Miguel pointed out a couple of mails ago the answer is known and
can be found in the handbook of Mathematical Functions if you want an
aswer in terms of psi and polygamma functions (do these exist in maxima?).

But then one still needs to implement properties of these functions
to simplify further. So it is perhaps better to use a special algorithm
which has these simplifications built in (when they apply).
This is what the paper by Ash and Catou does.

But I don't really care as long as maxima would be able to evaluate
something like sum(1/(3*n+1)^2+1/(3*n+2)^2,n,1) (I assume macsyma can do
that?).

Regards,
Michel