I have the following summation: \begin{equation}\sum_{n=2}^\infty c_n(n)(n-1)(x-1)^{n-2}+(x+1)\sum_{n=1}^\infty nc_n(x-1)^{n-1}+\sum_{n=1}^\infty nc_n(x-1)^{n-1}+(x+1)^2\sum_{n=0}^\infty c_n(x-1)^n +2(x-1)\sum_{n=0}^\infty c_n(x-1)^n+\sum_{n=0}^\infty c_n(x-1)^n \end{equation} Note the incline is bad on this post I will try to fix, after I post it buts just one large sum. $$\newcommand{\sm}[2]{\sum_{#1}^{#2}}$$ I managed to combine them using this post.
Which I managed to convert to the following: \begin{equation}\sm{k=0}{\infty}(c_{k+2}(k+2)(k+1)+kc_k+(k+1)c_{k+1}+c_k)(x-1)^k+ \bbox[5px,border:2px solid red]{\sm{n=0}{\infty}c_n(x-1)^{n+2}+\sm{n=0}{\infty}2c_n(x-1)^{n+1}} \end{equation} What's left is these two odd balls as I call them. I am trying to combine them into that nice summation with k, because the work is for a power series solution to a DE. My question is, is there a method to combine them.
$$l=\sum_{n=0}^{\infty}c_n(x-1)^{n+2}+\sum_{n=0}^{\infty}2c_n(x-1)^{n+1}$$ Rewrite this part as: $$l=\sum_{n=2}^{\infty}c_{n-2}(x-1)^{n}+\sum_{n=1}^{\infty}2c_{n-1}(x-1)^{n}$$ $$l=2c_0(x-1)+\sum_{n=2}^{\infty}c_{n-2}(x-1)^{n}+\sum_{n=2}^{\infty}2c_{n-1}(x-1)^{n}$$ $$l=2c_0(x-1)+\sum_{n=2}^{\infty}(c_{n-2}+2c_{n-1})(x-1)^{n}$$ Then start all the others sums at indice $n=2$