It's fairly easy to see that the series $$1-\tfrac{1}{3}+\tfrac{1}{5}-\cdots=\tfrac{1}{4}\pi$$ is : 1. Convergent to the value given, and - 2. Very slowly converging, which can be seen just by testing with a calculator. Euler had some methods to turn certain slowly convergent series into more quickly convergent series - applying Euler's method to the above gives the series $$\sum\limits_{n=0}^{\infty}\frac{1}{2^{n+1}}\sum\limits_{k=0}^{n}\binom{n}{k}\frac{(-1)^k}{2k+1}=\sum\limits_{n=0}^{\infty}\frac{(2n)!!}{2^{n+1}(2n+1)!!}=\tfrac{1}{4}\pi$$ which converges much more quickly. What I'm interested in is if your aim was to find a rapidly convergent series right from the get-go, what are the methods for going about this? For example, compare with the 'well-known' formula


which gives $\pi\simeq\tfrac{9801}{2206\sqrt{2}}=3.141592\dots$ after just the $n=0$ step. So, what I'm asking is - if you are searching for a rapidly convergent series, what qualities do you typically want your result to have, and how do you go about finding one? For example, are there certain 'starting points' that typically give rapid convergence - e.g. if a series is derived via. a hypergeometric function identity (like I believe the second formula above is), for example, does it tend to be rapidly convergent? If so, does the same happen for the gamma function, the circular functions, etc. and why?


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