A friend of mine studies mathematics and he told me about a project he has to do and we worked on it together a bit: Numerically calculate the arctan.
He dug out a nice series by (the one and only) Euler and we started massaging the thing to get it into a bit of a nicer form (there were (n!)^2 and other shenanigans) and we eventually succeeded after some stupid simple errors and arrived at a quite simple recursive progression. After that he also found a formula to transform a given value into the region where our formula actually mimics the arctan and we proceeded to proof this formula. The programming was straight forward and now we only have to find the radius of convergence which I suspect is pi^2 (but no proof).
I had a lot of fun doing this, fiddling around with the formulas and then programming it to see it actually becoming real.