>
> 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.
>
Perhaps rational roots are also ok. E.g. it is easy to express
sum(1/(2k+1)^2,k,0,inf) in terms of zeta(2). If I remember correctly
from my number theory courses this procedure may generalize (certain
gauss sums appear).
Michel