- #1
crazycool2
- 16
- 0
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
\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 xn 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 xn 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?
\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 xn 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 xn 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?