Ramanujan's fastest series for 1/π(PI)

In his paper Modular equations and approximations to $\pi$ Ramanujan gave the following spectacular series for $1/\pi$: $$\frac{1}{2\pi\sqrt{2}} = \frac{1103}{99^{2}} + \frac{27493}{99^{6}}\frac{1}{2}\frac{1\cdot 3}{4^{2}} + \frac{53883}{99^{10}}\frac{1\cdot 3}{2\cdot 4}\frac{1\cdot 3\cdot 5\cdot 7}{4^{2}\cdot 8^{2}} + \cdots\tag{1}$$ without any proof. The series converges very fast with each term giving almost 8 digits of the final sum. It was proved by Borwein brothers in 1987 with some calculations assisted via computer software. And most recently in 2025 it was proved by Jesus Guillera using MAPLE software. There is no known proof using hand calculation available so far. Using ideas from a paper by Pablo F. Refolio, an independent researcher from Spain, I will present a proof which involves calculations via hand, pen and paper.

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Ramanujan's take on Chudnovsky series for 1/π(PI): Part 2

In the previous post we have handled the evaluation of $P_n=P(-e^{-\pi\sqrt{n}}) $ for $n=11,27$. We will evaluate $P_n$ for $n=19,35$ in the current post and also discuss an empirical approach for $n=43,67,163$. Finally we will use the information in table given by Ramanujan to obtain certain series for $1/\pi$ (including the famous one by Chudnovsky brothers).

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Ramanujan's take on Chudnovsky series for 1/π(PI): Part 1

We have discussed a proof of Chudnovsky series for $1/\pi$ in this post based on Ramanujan's ideas as presented in this post. However a serious look at one of the pages from his lost notebook suggests that Ramanujan used a slightly different approach to obtain Chudnovsky type series and he also performed all the desired calculations needed to get the series in explicit form. This is what we intend to discuss in the current post.

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