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

canseeker
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2 Answers2

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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.

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

Lutz Lehmann
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    Wonderful argument. Note that this can also be expressed in the form $\frac {\sqrt{2 \pi}}{2} e^{x^2/2}erf(\frac{x}{\sqrt{2}})$ – infinitylord Jan 11 '17 at 20:05
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Look at the series expansions for the Error function

amWhy
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    No, those have complete factorials in the denominators. – hmakholm left over Monica Jan 11 '17 at 19:38
  • You are right. Errorfunctions a better hint? –  Jan 11 '17 at 19:50
  • Ok, so now this is where I say "Please see identity $(9-10)$ in the following link" – Simply Beautiful Art Jan 11 '17 at 19:52
  • @SimpleArt Please don't condone such an answer, nor attempt to translate and edit it into a form the answerer probably doesn't understand (but has googled) and in doing so, essentially endorsing such actions. This is, and remains, a link only answer. If you feel you must intervene for the sake of "Truth" (with a capital "T), then vote for the correct answer, not one that even the asker could have written. – amWhy Jan 14 '17 at 18:53