If anyone's curious, that series comes from the equation Arctan(1)=Pi/4 and the Taylor expansion around zero of Arctan(x) = x - x^3/3 + x^5/5 + ... where x=1.

If anyone's curious, that series comes from the equation Arctan(1)=Pi/4 and the Taylor expansion around zero of Arctan(x) = x - x^3/3 + x^5/5 + ... where x=1. What makes this expansion particularly painful is that usually Taylor expansion is done around a point where x is small, such that the x^n numerators of subsequent terms get progressively smaller. But in this case the numerator is always 1, so the series depends entirely on the denominator for convergence. Meanwhile, the denominator doesn't grow by the usual increasing factorial terms common to Taylor series, but grows only linearly. So both effects combined cause this expansion to take a very long time to converge.