I have the following start to an example:
$(1-x^2)y''-2xy'+\ell(\ell+1)y=0$ ;
$y= \sum_{n=0}^\infty C_nx^{n+r}$
Step 1:(Subbing in y'',y' and y) $(1-x^2)[\sum_{n=0}^\infty C_n(n+r)(n+r-1)x^{n+r-2}]-2x[\sum_{n=0}^\infty C_n(n+r)x^{n+r-1}]+\ell(\ell+1)[\sum_{n=0}^\infty C_nx^{n+r}=0$
This is then simplified to:
$\sum_{n=0}^\infty C_n(n+r)(n+r-1)x^{n+r-2}-\sum_{n=0}^\infty C_n(n+r)(n+r-1)x^{n+r}$
$\bbox[yellow]{-2\sum_{n=0}^\infty C_n(n+r)x^{n+r}}+\ell(\ell+1)\sum_{n=0}^\infty C_nx^{n+r}=0$
Then the like terms are further simplified to give:
$\sum_{n=0}^\infty C_n(n+r)(n+r-1)x^{n+r-2}+\sum_{n=0}^\infty x^{n+r}C_n[\ell(\ell+1)-(n+r)(n+r+1)]=0$
However, I fail to see why the $-2\sum_{n=0}^\infty C_n(n+r)x^{n+r}$ is not present in the final equation. Why has it disappeared? I've seen this happen in a few examples now but don't understand why it is okay to take it away.
Any insight would be appreciated.
It hasn't disappeared. It was combined with the $-\sum_{n=0}^\infty C_n (n+r)(n+r-1) x^{n+r}$. $$ - (n+r)(n+r-1) - 2 (n+r) = -(n+r)(n+r+1)$$