Ramanujan's continued fractions

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Ramanujan's continued fractions are a series of interesting closed-form expressions for non-simple continued fractions developed by Indian mathematician Srinivasa Ramanujan.

Contents

[edit] Examples

Among the expressions developed by Ramanujan are two which are nearly equal to one:

[edit] Nearly one

{1\over 1+{e^{-2\pi}\over 1+{e^{-4\pi}\over 1+\dots}}} = \left({\sqrt{5+\sqrt{5}\over 2}-{\sqrt{5}+1\over 2}}\right)e^{2\pi/5} = e^{2\pi/5}\left({\sqrt{\phi\sqrt{5}}-\phi}\right) = 0.9981360\dots

where φ is the golden ratio (Approximately 1.618)

The multiplicative inverse of this expression is:

1 + {e^{-2\pi}\over 1+{e^{-4\pi}\over 1+{e^{-6\pi}\over 1+\dots}}}=\frac{1}{2}\left[1+\sqrt{5}+\sqrt{2(5+\sqrt{5})}\right]\,e^{-2\pi/5}
= \frac{e^{-2\pi/5}}{\sqrt{\phi\sqrt{5}}-{\phi}}=1.0018674...

[edit] Even closer to one

{1\over 1+{e^{-2\pi\sqrt{5}}\over 1+{e^{-4\pi\sqrt{5}}\over 1+\dots}}}
=\left(\frac{\sqrt{5}}{1+[5^{3/4}(\phi-1)^{5/2}-1]^{1/5}}-{\phi}\right)\,e^{2\pi/\sqrt{5}}=0.99999920...

The multiplicative inverse of this expression is:

1 + {e^{-2\pi\sqrt{5}}\over 1+{e^{-4\pi\sqrt{5}}\over 1+\dots}}
=\frac{e^{-2\pi/\sqrt{5}}}{\frac{\sqrt{5}}{1+\left[5^{3/4}(\phi-1)^{5/2}-1\right]^{1/5}}-{\phi}}=1.000000791267...

[edit] References

[1]