1
$\begingroup$

In a letter to G.H Hardy on January $13^{\text{th}}$, Ramanujan posted this interesting continued fraction:$$\cfrac {1}{1+\cfrac {e^{-2\pi}}{1+\cfrac {e^{-4\pi}}{1+\cfrac {e^{-6\pi}}{1+\cfrac {e^{-8\pi}}{1+\ddots}}}}}=\left(\sqrt{\dfrac {5+\sqrt5}2}-\dfrac {1+\sqrt5}2\right)\sqrt[5]{e^{2\pi}}\tag1$$ I find this interesting that an infinite continued fraction can be represented by a finite nested radical!

Question:

  1. Is there a way to generate similar identities to $(1)$?
  2. How did Ramanujan come up with $(1)$ in the first place?
$\endgroup$
  • 4
    $\begingroup$ Identity (1) is immediately obvious to the most casual observer. $$ $$As long as that observer is Ramanujan. $\endgroup$ – Mark Fischler Jan 18 '17 at 19:28
  • 1
    $\begingroup$ See Rogers-Ramanujan continued fraction. $\endgroup$ – Robert Israel Jan 18 '17 at 19:35
  • $\begingroup$ ... and the references there. In particular, you might look at Berndt's book "Ramanujan's Notebooks: Part III". $\endgroup$ – Robert Israel Jan 18 '17 at 19:40
  • $\begingroup$ The fact that it is on January 13th or February 29th without knowing the year isn't that important, don't you think :) ? $\endgroup$ – Jean Marie Jan 18 '17 at 19:59
  • $\begingroup$ @RobertIsrael Do you happen to know what page in Ramanujan's Notebook? $\endgroup$ – Frank Jan 18 '17 at 21:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.