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I'm searching for a proof of one impressive Ramanujan result. Not one in particular, the only request I have is to be really impressive. For example $$ \sqrt{\phi+2}-\phi=\frac{e^{-2\pi/5}}{1+\frac{e^{-2\pi}}{1+\frac{e^{-4\pi}}{1+\cdots}}} $$ where $\phi=\frac{1+\sqrt5}{2}$.

Or maybe $$ \frac1{\pi}=\frac{2\sqrt2}{9801}\sum_{n=0}^{+\infty}\frac{(4n)!(1103+26390n)}{(n!)^4396^{4n}}\;. $$

Can someone suggest me a precise reference where to find such a proof?

Thank you all

Joe
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    The first result is known as Rogers-Ramanujan continued fraction. In the reference section of the wiki page, there is a survey article of this stuff by Berndt, B, et al. – achille hui Jul 27 '14 at 13:20
  • Both the results you have mentioned are very famous. First one is proved in one of my blog posts http://paramanands.blogspot.com/2013/09/values-of-rogers-ramanujan-continued-fraction-part-1.html and I have given Ramanujan's theory of series for $1/\pi$ in posts starting from http://paramanands.blogspot.com/2012/03/modular-equations-and-approximations-to-pi-part-1.html – Paramanand Singh Oct 03 '16 at 10:35

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The series for $1/\pi$ is proved in J. M. Borwein and P. B. Borwein, Pi and the AGM; A Study in Analytic Number Theory and Computational Complexity, Wiley, New York, 1987.

See also Motivation for Ramanujan's mysterious $\pi$ formula

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Gerry Myerson
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