According to WolframMathWorld (formulae $(9)$ and $(10)$ there) the Error Function may be expanded for $x\ll1$ by
$$\operatorname{erf}(x)~=~\frac{e^{-x^2}}{\sqrt\pi}\sum_{n\geqslant1}\frac{2^n}{(2n-1)!!}x^{2n-1}\tag1$$
(Note that on the MathWorld the given formulaes, $(9)$ and $(10)$, do not match, but WolframAlpha agrees on the version I gave. I think it is just a typo on MathWorld)
As it turned out this one can be rather useful, so I wanted to know how one can derive such a result. However, I have no idea where to start or how to actually search for a proof ("proof of formula $(9)$ on the MathWorld site of the Error Function" does not seem to be a good key word, though ^^).
I know how to derive the classical series expansion of the Error Function via termwise integration but this series does not seem to be of any help here.
To put it straight: how to prove $(1)$? Is there a approximation of $e^{-x^2}$ only working for $x\ll1$ which allows us to introduce this alternative series?
Thanks in advance!
Write $$\operatorname{erf}(x)=\frac2{\sqrt{\pi}}e^{-x^2}f(x)$$ for some odd function $f$. Differentiate and multiply by $e^{x^2}$ to find $f’(x)-2xf(x)=1$. Expand $f$ as $$f(x)=\sum_{n=1}^{\infty}c_nx^{2n-1}.$$ Then the coefficients satisfy the recursion $$c_1=1,\quad c_{n+1}=\frac{2}{2n+1}c_n.$$