Editing Iterated function (section)

====Example 2====
Find the value of <math>\sqrt{2}^{ \sqrt{2}^{\sqrt{2}^{\cdots}} }</math> where this is done ''n'' times (and possibly the interpolated values when ''n'' is not an integer). We have {{math|1=''f''(''x'') = {{sqrt|2}}<sup>''x''</sup>}}. A fixed point is {{math|1=''a'' = ''f''(2) = 2}}.

So set {{math|1=''x'' = 1}} and {{math|''f'' <sup>''n''</sup> (1)}} expanded around the fixed point value of 2 is then an infinite series,
<math display="block">
\sqrt{2}^{ \sqrt{2}^{\sqrt{2}^{\cdots}} } = f^n(1) = 2 - (\ln 2)^n + \frac{(\ln 2)^{n+1}((\ln 2)^n-1)}{4(\ln 2-1)} - \cdots
</math>
which, taking just the first three terms, is correct to the first decimal place when ''n'' is positive. Also see [[Tetration]]: {{math|1=''f'' <sup>''n''</sup>(1) = <sup>''n''</sup>{{sqrt|2}}}}. Using the other fixed point {{math|1=''a'' = ''f''(4) {{=}} 4}} causes the series to diverge.

For {{math|1= ''n'' = −1}}, the series computes the inverse function {{sfrac|2|ln ''x''|ln 2}}.