find the sum of the following series using Maclaurins expansion

Question

Find the sum of the following series: $$\sum_{n=0}^\infty {x^{n}}{\sinh(5n+5)}$$

The sum for $ {\sinh(5n+5)}$ is as it follows

$$\sum_{n=0}^\infty \frac{(5n+5)^{2n+1}}{(2n+1)!}$$

And now I do not know how to continue to find this sum of series , can anyone help me .

Thank you all !

Answer 1

Assuming that $|x|<\frac{1}{e^5}$ (otherwise the series is divergent) you just have a geometric series:

$$\sum_{n\geq 0} x^n \sinh(5n+5) = \frac{1}{2}\left(\frac{e^5}{1-e^5 x}-\frac{e^{-5}}{1-e^{-5}x}\right)=\frac{e^{10}-1}{2(e^5-x)(e^5 x-1)}.$$