Hi, I'm working with series solutions of differential equations and I have come across something that has troubled me other courses as well. given that(adsbygoogle = window.adsbygoogle || []).push({});

\begin{equation}

\sum_{n=0}^{\infty} c_{n+2}x^n+e^{-x} \sum_{n=0}^{\infty}c_{n}x^n \\

\text{where}\\

e^{-x}=\frac{1}{\sum_{n=0}^{\infty}\frac{x^n}{n!}}\\

\sum_{n=0}^{\infty} c_{n+2}x^n+\frac{ \sum_{n=0}^{\infty}c_{n}x^n }{\sum_{n=0}^{\infty}\frac{x^n}{n!}}

\end{equation}

now my problem is I have the x^{n}in every term and the limits are the same, but I have a fraction of sums and I want to a way to make it simpler. can the x^{n}cancel each other in the fraction. if so

i then have

\begin{equation}

\sum_{n=0}^{\infty} c_{n+2}x^n+ \sum_{n=0}^{\infty} c_n*n!

\end{equation}

Is this even allowed?

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# I Summation algebra

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