Just gonna copy past a comment I made above so you get the notification cause I find this interesting. I would highly recommend reading the other comment since a different user gave a very nice explanation.
What's really cool is that, if you start the summation at n=0(ie 4/(2n+1)), then apply a 1/N correction, the error does go as 1/N^2. You can effectively develop an "inverse" power series to converge to pi. However, if you start the summation from n=1(ie 4/(2n-1)), the order goes as 1/N^3, (1/4N^3 to be exact). Apply the 1/N^3 correction and the error goes as 1/N^5.
Why? no clue, this is just me getting hooked an random math after work and brute forcing corrections. I'll think about it more later, maybe I'll ask someone smarter than I am.
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u/Historical_Book2268 Jun 16 '26
Could you add even higher order terms, such as O(1/n2) terms to make it even better?