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 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 A to 2 decimal places:

1 A = 2 2 9801 ⋅ ∑ k = 0 ∞ ( 4 k ) ! ⋅ ( 1103 + 26390 k ) ( k ! ) 4 ⋅ 39 6 4 k \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}}}


The answer is 3.14.

Guillermo Templado by Guillermo Templado , 46, Spain

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1 solution

Rushikesh Jogdand
May 17, 2016

1 π = 2 2 9801 ⋅ ∑ k = 0 ∞ ( 4 k ) ! ⋅ ( 1103 + 26390 k ) ( k ! ) 4 ⋅ 39 6 4 k \displaystyle \large {\frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \cdot \sum_{ k = 0}^\infty \frac{(4k)! \cdot (1103 +26390k)}{(k!)^{4}\cdot 396^{4k}}} This formula by Ramaujan could calculate value of π \pi to much more preision with small value of parameter than any other contemporary formula.

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Sorry, I don't agree. Firstly, this is not a proof, and secondly there are derivated formulas of this one which let to calculate 16 digits of π \pi every iteration and this formula only lets 8 digits of π \pi each iteration...

Guillermo Templado - 5 years ago

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Well I did say contemporary .

Rushikesh Jogdand - 5 years ago

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