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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?
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    Identity (1) is immediately obvious to the most casual observer. $$ $$As long as that observer is Ramanujan. – 2017-01-18
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    See [Rogers-Ramanujan continued fraction](https://en.wikipedia.org/wiki/Rogers%E2%80%93Ramanujan_continued_fraction). – 2017-01-18
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    ... and the references there. In particular, you might look at Berndt's book "Ramanujan's Notebooks: Part III". – 2017-01-18
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    The fact that it is on January 13th or February 29th without knowing the year isn't that important, don't you think :) ? – 2017-01-18
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    @RobertIsrael Do you happen to know what page in Ramanujan's Notebook? – 2017-01-18
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    In particular, page 79. But much of chapter 16 will be relevant. – 2017-01-18

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