> I apologize, Levin can also do this! Now it seems to me that levin is better.
>
> - M
>
>
>
Hmm I tried levin_u_sum(1/(n^2+x),n,1,10) and got a very complicated
rational function in x.
Do you mean sum(1/(n^2+n),n,1,inf)? Yes levin can do this. On the other
hand
sum(1/n^2-1/(n+1)^2,n,1,inf) does not work (symbolically I mean). It
does not seem to work with ratsum either.
But I suppose zeilberger or gosper can do it trivially.
I think maxima is sorely lacking in symbolic summation capabilities
(compared to Maple or Mathematica). For example it does
not seem to know that sum(1/n!,n,1,inf) is equal to %e. I didn't try it
but I assume zeilberger does not help with this
sum since the partial sums have no closed expression (I am not very
familiar with zeilberger).
I think we should start implementing some symbolic sums. But I am not an
expert in this.
It seems to me that sum(p(n)/q(n),n,1,inf) where q has only real integer
roots can be summed symbolically by splitting p(n)/q(n) in
partial fractions. The result would be in terms of integer values of
the zeta function.
Do you know more general situations? A while ago Jack D'Aurizio of the
University of Pisa, Italy sent me the following for
symbolically summing sum(1/(n^2+1),n,1,inf):
> Maybe useless, but in order to sum
> sum(1/(1+n^2),n,1,inf)
> one can consider that
> sin(pi z)/(pi z) = prod(n=1..inf) (1 - x^2 / n^2)
> sinh(pi z)/(pi z) = prod(n=1..inf) (1 + x^2 / n^2)
> (weierstrass products) and compute logartithmic derivatives
I have not thought about this yet.
Regards,
Michel