A proof of today's @AnalysisFact tweet.
The series expansion for arctan(x) & an infinite series for π.pic.twitter.com/twTXKGtIay
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Extra Credit: does the series converge on the endpoints x=-1,1? If so, what other (perhaps more elegant) infinite series can we write down for π?
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Absolutely yes. 4*nsum(lambda n:\ (-1)**n/(2*n+1), (0,inf)) >> 3.14159265358979 And it's called as the Gregory series, by a special case for the series expansion of arctangent function that Leibniz has proposed. The value for arctan({1,-1}) converges to π/4,-π/4, respectively.
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