Ramanujan Misterious PI formula..

tpm

Is there any mathematical explanation to the incredible fast converging formula by Ramanujan?:

[tex] \frac{2\sqrt{2}}{9801} \sum^\infty_{k=0} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}=\frac{1}{\pi}[/tex]

or simply "ocurred to him" and put it on a paper.

Gib Z

There most likely is a mathematical explanation to the series, but from arguments that are far more advanced than my knowledge.

There is however a very small chance it just luckily occurred to him, just as this interesting approximation did (he got it in a dream apparently) : [tex]\sqrt[4]{\frac{2143}{22}}[/tex]

Thats accurate to 9 digits, and came from a dream with no mathematical basis, so obviously Ramanujan was extremely proficient in his numeracy.

I can only offer 2 ideas :

The first is the following expression for pi, which looks like it may be somehow related to the series and had been transformed :

[tex] \frac{\sqrt2}2 \cdot \frac{\sqrt{2+\sqrt2}}2 \cdot \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2 \cdot \cdots = \frac2\pi[/tex]

The 2nd idea is to send an email to the Chudnovsky brothers, because I know that the series you ask about is in fact the basis for this faster series:

[tex]\frac{1}{\pi} = 12 \sum^\infty_{k=0} \frac{(-1)^k (6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 640320^{3k + 3/2}}[/tex]

Maybe they can help you.

Gib Z

O just in case there was any confusion over my last part of the post, the Chudnovsky brothers discovered that series.