I am trying to change a double sum limits but I am not sure if I am doing correctly. The sum is and my solution is
$\sum_{r=0}^n \sum_{m=0}^\infty \frac{a^m\sqrt{(m+r)!}}{m!}=\sum_{r=0}^{m'}\sum_{m'=0}^\infty \frac{a^{m'-r}\sqrt{m'!}}{(m'-r)!};\quad with\quad m'=m+r$
Currently, I am struggling to find the solution to an equation of which this double sum is part, and I think this double sum could be the problem. So I would be really grateful if some could help me to understand if my change of limits is correct. Thanks in advance.
For $m'=m+r$ we have that
$$\sum_{r=0}^n \sum_{m=0}^\infty \frac{a^m\sqrt{(m+r)!}}{m!}=\sum_{r=0}^{n}\sum_{m'=r}^\infty \frac{a^{m'-r}\sqrt{m'!}}{(m'-r)!}$$