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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 !

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  • $\begingroup$ Hint: write $\sinh $ in terms of $\exp $ ans see what kind of series you get. $\endgroup$ – Matematleta Jun 14 '15 at 15:02
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Assuming that $|x|<\frac{1}{e^5}$ (otherwise the series is divergent) you just have a geometric series:

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

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    $\begingroup$ Completely clear , thank You very much for helping me ! :) $\endgroup$ – MATH14 Jun 14 '15 at 15:27

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