Comment #17 on issue 1028 by asmeurer: Sum().doit() doesn't do it http://code.google.com/p/sympy/issues/detail?id=1028
You are correct in that the summation algorithms are similar to the integration algorithms (for example, the Karr algorithm has been called the counterpart to the Risch algorithm for symbolic integration). But the way that it is implemented is completely different from the way that you would do it by hand, because those methods tend to be slower and less complete than the algorithmic methods.
For example, if you wanted to integrate a rational function by hand, you would probably use partial fractions to split the function apart. But this is actually very inefficient. The algorithmic way to do it involves a lot of polynomial division. It would be painful to do by hand, but computers can do it very fast and efficiently. Furthermore, partial fractions only works for a subset of rational functions, whereas the algorithmic method works for *any* rational function.
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