Hint needed to figure out sum of the series $x + \frac{x^3}{1\cdot 3} + \frac{x^5}{1\cdot 3\cdot 5}+...$

Question

$$x + \frac{x^3}{1\cdot 3} + \frac{x^5}{1\cdot 3\cdot 5}+...$$ I was wondering how should I move ahead to try to figure out the sum of this series. I will appreciate any hints.

Answer 1

Look at the series expansions for the Error function

Answer 2

Note that $f'(x)=1+x·f(x)$ with $f(0)=0$, so that one gets the series expression as solution to this initial value problem.

$$ \frac{d}{dx}(e^{-x^2/2}f(x))=e^{-x^2/2}\\~\\ e^{-x^2/2}f(x)-0=\int_0^xe^{-s^2/2}\,ds\\~\\ f(x)=\int_0^xe^{(x^2-s^2)/2}\,ds $$