Integral of a series

Question

I cannot solve this integral, can anyone help me?

$$\int_0^\infty \left(x^3 \sum_{n=1}^{+ \infty} e^{-nx} \right)dx$$

Thank you in advance

A related problem. – Mhenni Benghorbal Aug 20 '13 at 20:22 — Mhenni Benghorbal, Aug 20 '13 at 20:22

score 7 · Accepted Answer · 2013-08-20T08:45:18.150

7

By the change variable $t=nx$ we have $$\int_0^\infty x^3e^{-nx}dx=\frac{1}{n^4}\int_0^\infty t^3e^{-t}dt=\frac{\Gamma(4)}{n^4}$$ so $$\int_0^\infty (x^3 \sum_{1}^ \infty e^{-nx})dx=\Gamma(4) \sum_{1}^ \infty\frac{1}{n^4}=\Gamma(4)\zeta(4)=3!\times \frac{\pi^4}{90}$$

edited Aug 20 '13 at 08:45

answered Aug 20 '13 at 08:32

Could you please specify the integral in the variable t? – Chaos Aug 20 '13 at 08:42
Yes I'll edit the answer. – Aug 20 '13 at 08:43
Thank you very much! How did you calculate the value of the series $\sum_{n=1}^{\infty}\frac{1}{n^4}$? – Chaos Aug 20 '13 at 08:53
You can calculate the Fourier series of the periodic function $|x|$ on the interval $[-\pi,\pi]$ and then apply the Parseval theorem. – Aug 20 '13 at 08:57
or using complex analysis or bernoulli numbers .. – what'sup Aug 20 '13 at 09:17

Integral of a series

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