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I have the sum

$$S=\frac 12 \sin x$$

and need to find the original function to this summation equation:

$$\sum_{n=1}^\infty f(x)=S$$

I am using the math software Maple, but can't solve this equation there. Is it possible to find the (or an) original function $f(x)$?

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You need to specify how $f(x)$ depends on $n$. If there is no dependence on $n$, as your notation indicate, then there is no solution. – Per Manne Nov 16 '12 at 15:35
up vote 5 down vote accepted

You have this formula that you can use: $$ \sin(x) = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}x^{2n+1}. $$ Now of course just put a $\frac{1}{2}$ in front. $$\begin{align} \frac{1}{2}\sin(x) &= \sum_{n=0}^\infty \frac{(-1)^n}{2(2n+1)!}x^{2n+1} \\ &= \sum_{n=1}^\infty \frac{(-1)^{n-1}}{2(2n-1)!}x^{2n-1}. \end{align} $$

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