I have the following: $$\left(\frac{C_0}{0!}a_m+\frac{C_1}{1!}a_{m-1}+...+\frac{C_m}{m!}a_0\right)x^m+\left(\frac{C_1}{1!}a_m+\frac{C_2}{2!}a_{m-1}+...+\frac{C_{m+1}}{(m+1)!}a_0\right)x^{m+1}+...+\left(\frac{C_i}{i!}a_m+\frac{C_{i+1}}{(i+1)!}a_{m-1}+...+\frac{C_{m+i}}{(m+i)!}a_0\right)x^{m+i}+...$$
I am trying to write this as a summation of the form $$\sum_{n=m}^{\infty}\alpha x^n$$ where $\alpha $ is just some expression containing the fixed number summation above. I think I can write this so far as
$$\sum_{n=m}^{\infty}\left(\frac{C_{n-m}}{(n-m)!}a_m+\frac{C_{n+1-m}}{(n+1-m)!}a_{m-1}+...+\frac{C_{n+m-m}}{(n+m-m)!}a_0\right)x^n$$
Maybe it's just because I'm tired but how can I write the inside series as a summation? it is a fixed number of summands and the $a$ terms always start at $a_m$ and end with $a_0$
Is it simply
$$\sum_{n=m}^\infty \sum_{k=0}^m \frac{C_{n+k-m}}{(n+k-m)!}a_{m-k}x^n$$
Or are my indices / bounds incorrect?