Changing the limits of a power series

Question

I don't know how to change the limits of the summations of the power series. Am I allowed to rewrite the $$ \sum_{i=0}^pa_i\,r^{i+4}-\sum_{i=2}^pa_i\,r^{i+2}+\sum_{i=4}^pa_i\,r^i-\sum_{i=3}^pa_i\,r^{i+1}=0 \tag{1} $$ as $$ \sum_{i=4}^pa_{i-4}\,r^{i}-\sum_{i=4}^pa_{i-2}\,r^{i}+\sum_{i=4}^pa_i\,r^i-\sum_{i=4}^pa_{i-1}\,r^{i}=0 \tag{2} $$ ?? Note that the upper limit of summations is not $\infty$

Answer 1

Note that $$\sum_{i=0}^pa_ir^{i+4}=a_0r^4+\ldots+a_pr^{p+4}$$ and $$\sum_{i=4}^pa_{i-4}r^{i}=a_0r^4+\ldots+a_{p-4}r^{p}$$ so you want to adjust the top limit from $p$ to $p+4$ (and accordingly for the other parts).

Once you have done that, collecting like powers is easy: $$\begin{align}0 &=\sum_{i=0}^pa_i\,r^{i+4}-\sum_{i=2}^pa_i\,r^{i+2}+\sum_{i=4}^pa_i\,r^i-\sum_{i=3}^pa_i\,r^{i+1}\\ &=\sum_{i=4}^{p+4}a_{i-4}\,r^{i}-\sum_{i=4}^{p+2}a_{i-2}\,r^{i}+\sum_{i=4}^pa_i\,r^i-\sum_{i=4}^{p+1}a_{i-1}\,r^{i}\\ &=\sum_{i=4}^p\left(a_{i-4}\,r^{i}-a_{i-2}\,r^{i}+a_i\,r^i-a_{i-1}\,r^{i}\right)\\&\qquad+(a_{p-3}r^{p+1}+a_{p-2}r^{p+2}+a_{p-1}r^{p+3}+a_pr^{p+4})\\&\qquad-(a_{p-1}r^{p+1}+a_pr^{p+2})-a_pr^{p+1}\\ &=\sum_{i=4}^p(a_{i-4}-a_{i-2}+a_i-a_{i-1})r^i\\&\qquad+(a_{p-3}-a_{p-1}-a_p)r^{p+1}+(a_{p-2}-a_p)r^{p+2}+a_{p-1}r^{p+3}+a_pr^{p+4}\end{align} $$