Editing Error function (section)

===Asymptotic expansion===
A useful [[asymptotic expansion]] of the complementary error function (and therefore also of the error function) for large real {{mvar|x}} is
<math display="block">\begin{align}
\operatorname{erfc} x &= \frac{e^{-x^2}}{x\sqrt{\pi}}\left(1 + \sum_{n=1}^\infty (-1)^n \frac{1\cdot3\cdot5\cdots(2n - 1)}{\left(2x^2\right)^n}\right) \\[6pt] 
&= \frac{e^{-x^2}}{x\sqrt{\pi}}\sum_{n=0}^\infty (-1)^n \frac{(2n - 1)!!}{\left(2x^2\right)^n},
\end{align}</math>
where {{math|(2''n'' − 1)!!}} is the [[double factorial]] of {{math|(2''n'' − 1)}}, which is the product of all odd numbers up to {{math|(2''n'' − 1)}}. This series diverges for every finite {{mvar|x}}, and its meaning as asymptotic expansion is that for any integer {{math|''N'' ≥ 1}} one has
<math display="block">\operatorname{erfc} x = \frac{e^{-x^2}}{x\sqrt{\pi}}\sum_{n=0}^{N-1} (-1)^n \frac{(2n - 1)!!}{\left(2x^2\right)^n} + R_N(x)</math>
where the remainder is
<math display="block">R_N(x) := \frac{(-1)^N \, (2 N - 1)!!}{\sqrt{\pi} \cdot 2^{N - 1}} \int_x^\infty t^{-2N}e^{-t^2}\,\mathrm dt,</math>
which follows easily by induction, writing
<math display="block">e^{-t^2} = -\frac{1}{2 t} \, \frac{\mathrm{d}}{\mathrm{d}t} e^{-t^2}</math>
and integrating by parts.

The asymptotic behavior of the remainder term, in [[Landau notation]], is
<math display="block">R_N(x) = O\left(x^{- (1 + 2N)} e^{-x^2}\right)</math>
as {{math|''x'' → ∞}}. This can be found by 
<math display="block">R_N(x) \propto \int_x^\infty t^{-2N}e^{-t^2}\,\mathrm dt = e^{-x^2} \int_0^\infty (t+x)^{-2N}e^{-t^2-2tx}\,\mathrm dt\leq e^{-x^2} \int_0^\infty x^{-2N} e^{-2tx}\,\mathrm dt \propto x^{-(1+2N)}e^{-x^2}.</math>
For large enough values of {{mvar|x}}, only the first few terms of this asymptotic expansion are needed to obtain a good approximation of {{math|erfc ''x''}} (while for not too large values of {{mvar|x}}, the above Taylor expansion at 0 provides a very fast convergence).