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 !

  • $\begingroup$ Hint: write $\sinh $ in terms of $\exp $ ans see what kind of series you get. $\endgroup$ – Matematleta Jun 14 '15 at 15:02

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

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

  • 1
    $\begingroup$ Completely clear , thank You very much for helping me ! :) $\endgroup$ – MATH14 Jun 14 '15 at 15:27

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.