>>10952383That series os for 4*arctan(1).
Arctan(1) = pi/4 (look at a trig circle).
If uv != 1 then arctan((u+v)/(1-uv)) = arctan(u) + arctan(v) and you can use this identity to create lots of other formulas for pi based on the arctan series.
This was how it was done in the 18th century, Euler computed pi to 300+ digits using one such formula and someone would compute pi up to some more digits every once in a while simply by reapplying the identity again and again.
There's also several convergence acceleration methods you could use to turn that series into another that also goes to pi.
But that's transforming the series you already have.
Thanks to a result due to Euler, we know that = pi^2/6. So just multiply that by 6 and take a square root.
There areany other identities.
Archimedes method was aproximating a circle with other polygons, you can get a sequence aproximating pi pretty fast.
Lots of modern methods get pretty crazy too.
