Mathematical explanation for Ramanujan's pi formula

[tpm]

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

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

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) : $$\sqrt[4]{\frac{2143}{22}}$$

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 :

$$\frac{\sqrt2}2 \cdot \frac{\sqrt{2+\sqrt2}}2 \cdot \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2 \cdot \cdots = \frac2\pi$$

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:

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

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.