Ramanujan's continued fractions

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

Examples

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

Nearly one

:$\{1over\; 1+\{e^\{-2pi\}over\; 1+\{e^\{-4pi\}over\; 1+dots\}\; =\; left(\{sqrt\{5+sqrt\{5\}over\; 2\}-\{sqrt\{5\}+1over\; 2\; ight)e^\{2pi/5\}\; =\; e^\{2pi/5\}left(\{sqrt\{phisqrt\{5-phi\}\; ight)\; =\; 0.9981360dots$where $phi$ is the golden ratio (Approximately 1.618)

The multiplicative inverse of this expression is:

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

Even closer to one

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

The multiplicative inverse of this expression is:

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

References

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