The man who knew infinity (Ramanujan)

Calculus Level 3

For this wonderful next formula given by Ramanujan around 1910 had to wait three quarters of a century to be demonstrated, but Ramanujan did not bother to do so. Bill Gosper a "hacker" used it to calculate seventeen million figures of \(A \). This formula has the surprising property of producing eight decimal each time is calculated with a more term. And you have a wonderful space for a proof... Ramanujan's formula has a closed beautiful form, submit \(A\) to 2 decimal places:

\[\displaystyle \large {\frac{1}{A} = \frac{2\sqrt{2}}{9801} \cdot \sum_{ k = 0}^\infty \frac{(4k)! \cdot (1103 +26390k)}{(k!)^{4}\cdot 396^{4k}}} \]

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