[ArbitraryLimits]

Not really related, but Gosper's algorithm (http://en.wikipedia.org/wiki/Gospers_algorithm) is the thing I've learned in the last few years that blew my mind the hardest. It's how Maple or Mathematica can reduce your crazy sums to a single formula. Earlier work was apparently pioneered by a nun: http://en.wikipedia.org/wiki/Mary_Celine_Fasenmyer