> There is a closed_form function for sum simplification in maxima. It
> uses Gosper and Zeilberger algorithms.
>
> (%i1) load(closed_form)$
> (%i2) sum(1/k/(k+1), k, 1, inf);
> (%o2) sum(1/(k*(k+1)),k,1,inf)
> (%i3) closed_form(%);
> (%o3) 1
>
> (%i6) sum(binomial(x,k)*binomial(y,n-k), k, 0, n);
> (%o6) sum(binom(x,k)*binom(y,n-k),k,0,n)
> (%i7) closed_form(%);
> Is x - n positive, negative, or zero? pos;
> Is y - n positive, negative, or zero? pos;
> (%o7) ((y+x)*(y+x-1)!)/(n!*(y+x-n)!)
> (%i8) factcomb(%);
> (%o8) (y+x)!/(n!*(y+x-n)!)
Great!!! I didn't know that. But I assume it only works when there is an
"anti-difference".
I tried
closed_form(sum(1/n!,n,0,inf));
but it didn't do anything (the answer is of course %e).
>
> The closed form of sum(1/(3*n+1)^2 + 1/(3*n+2)^2, n, 0, inf) is
> psi[1](1/3)/9 + psi[1](2/3)/9. Maxima does not compute it. Writing a
> program which computes the closed form solution of sums of rational
> function in terms of psi and polygamma should not be too hard. Since
> maxima can compute these it would be interesting.
>
> Andrej
The point is that there is a much simpler answer: namely 4/27*%pi^2.
I assume this can be deduced from the formal properties of the psi
function, but
maxima would have to know the appropriate simplification rules.
On the other hand it can easily deduced directly from the form of the sum
sum(1/(3*n+1)^2 + 1/(3*n+2)^2, n, 0, inf) that it is equal to
zeta(2)-zeta(2)/9.
Regards,
Michel