Exponential Function - Continued Fractions For ex

Continued Fractions For ex

A continued fraction for ex can be obtained via an identity of Euler:

\, \ e^x=1+\cfrac{x}{1-\cfrac{x}{x+2-\cfrac{2x}{x+3-\cfrac{3x}{x+4-\cfrac{4x}{x+5-\cfrac{5x}{x+6-\ddots}}}}}}

The following generalized continued fraction for e2x/y converges more quickly:

e^{2x/y} = 1+\cfrac{2x}{y-x+\cfrac{x^2}{3y+\cfrac{x^2}{5y+\cfrac{x^2}{7y+\cfrac{x^2}{9y+\cfrac{x^2}{11y+\cfrac{x^2}{13y+\ddots\,}}}}}}}

with a special case for x = y = 1:

e^2 = 7+\cfrac{2}{5+\cfrac{1}{7+\cfrac{1}{9+\cfrac{1}{11+\cfrac{1}{13+\ddots.}}}}}

Read more about this topic:  Exponential Function

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