Is the calculation of infinite sums solvable by a computer?

The question is: I give the computer a sum, such as $ \sum_{n=1}^\infty\frac{1}{n^3}$ , the computer is expected to return an elegant closed-form solution, because the answer may be irrational. Has this problem been solved using a computer? Or, has it been proved to be undecidable? Or is it open?