Editing Exponential function (section)

===Continued fractions===
The exponential function can also be computed with [[continued fraction]]s.

A continued fraction for {{math|''e''{{isup|''x''}}}} can be obtained via [[Euler's continued fraction formula|an identity of Euler]]:
<math display="block"> e^x = 1 + \cfrac{x}{1 - \cfrac{x}{x + 2 - \cfrac{2x}{x + 3 - \cfrac{3x}{x + 4 - \ddots}}}}</math>

The following [[generalized continued fraction]] for {{math|''e''{{isup|''z''}}}} converges more quickly:<ref name="Lorentzen_2008"/>
<math display="block"> e^z = 1 + \cfrac{2z}{2 - z + \cfrac{z^2}{6 + \cfrac{z^2}{10 + \cfrac{z^2}{14 + \ddots}}}}</math>

or, by applying the substitution {{math|1=''z'' = {{sfrac|''x''|''y''}}}}:
<math display="block">  e^\frac{x}{y} = 1 + \cfrac{2x}{2y - x + \cfrac{x^2} {6y + \cfrac{x^2} {10y + \cfrac{x^2} {14y + \ddots}}}}</math>
with a special case for {{math|1=''z'' = 2}}:
<math display="block">  e^2 = 1 + \cfrac{4}{0 + \cfrac{2^2}{6 + \cfrac{2^2}{10 + \cfrac{2^2}{14 + \ddots }}}} = 7 + \cfrac{2}{5 + \cfrac{1}{7 + \cfrac{1}{9 + \cfrac{1}{11 + \ddots }}}}</math>

This formula also converges, though more slowly, for {{math|''z'' > 2}}. For example:
<math display="block"> e^3 = 1 + \cfrac{6}{-1 + \cfrac{3^2}{6 + \cfrac{3^2}{10 + \cfrac{3^2}{14 + \ddots }}}} = 13 + \cfrac{54}{7 + \cfrac{9}{14 + \cfrac{9}{18 + \cfrac{9}{22 + \ddots }}}}</math>