This series converges even when x = 1 and since we know that arctan (1) = π/4, we then get the infinite series representation That's pretty cool. Unfortunately, it's not that efficient.

Moving beyond polygons The development of infinite series techniques in the 16th and 17th centuries greatly enhanced people’s ability to approximate pi more efficiently. An infinite series is ...

Really, all I have to do is change the value of pi in my program to e and run it again. And the best fractional representation of e is 566827 / 208524 that I found (I ran it up to numerator ...