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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The General Binomial Theorem: Part 2

In the previous post we established the general binomial theorem using Taylor's theorem which uses derivatives in a crucial manner. In this post we present another approach to the general binomial theorem by studying more about the properties of the binomial series itself. Needless to say, this approach requires some basic understanding about infinite series and we will assume that the reader is familiar with ideas of convergence/divergence of an infinite series and some of the tests for convergence of a series.

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