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I've been trying for quite a while to show that $$\int_0^\pi\left (\sum_{n=1}^\infty \frac{\sin(nx)}{n^3} \right) \, dx$$

$$=$$ $$2\sum_{n=1}^\infty \frac{1}{(2n-1)^4} $$ I can't quite seem to get it. It's in an intro Real Analysis, and we're learning about uniform convergence. I've worked with derivatives of series but not an integral. How can I go about solving this?

I know that $$\frac{d}{dx} \sum_{n=1}^\infty f_n (x) = \sum_{n=1}^\infty \frac{d}{dx}(f_n(x))$$ under the right conditions -- Can I use this with integrals? If so, how could I apply it?

Thanks

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To integrate a series term-by-term over a finite interval, it is sufficient that the series converges uniformly on the interval of integration. In this case we're fine, because $$ \left\lvert \frac{\sin{nx}}{n^3} \right\rvert \leqslant \frac{1}{n^3}, $$ which is a convergent series. Therefore we just have to compute the integral of each term, $$ \int_0^{\pi} \sum_{n=1}^{\infty} \frac{\sin{nx}}{n^3} \, dx = \sum_{n=1}^{\infty} \frac{1}{n^3} \int_0^{\pi} \sin{nx} \, dx, $$ and the last integral is $\frac{1}{n}(1-\cos{n\pi})$, which is $2/n$ if $n$ is odd and $0$ if $n$ is even. Reindexing the series then gives the result you want.

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  • $\begingroup$ I believe that the last integral should be $\frac{1-(-1)^n}n$, which is $\frac2n[n\text{ is odd}]$ where the brackets are Iverson Brackets. $\endgroup$ – robjohn Mar 26 '15 at 3:58
  • $\begingroup$ @robjohn D'oh, yes, forgot to do the actual integration... whoops. Thanks. Fixed now. $\endgroup$ – Chappers Mar 26 '15 at 4:01
  • $\begingroup$ Excellent answer, thank you $\endgroup$ – Jonathan Wu Mar 26 '15 at 23:59
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Show that the series converges uniformly on $[0,\pi]$ (use the Weierstrass $M$-test). This justifies integrating the series term-by-term over $[0,\pi]$. Then you compute

$$\sum_{n = 1}^\infty\frac{1}{n^3}\int_0^\pi \sin(nx)\, dx = \sum_{n = 1}^\infty \frac{1}{n^3}\cdot\frac{1 - \cos(n\pi)}{n} = \sum_{n = 1}^\infty \frac{1 - (-1)^n}{n^4} =2\sum_{n = 1}^\infty \frac{1}{(2n-1)^4}$$

since $1 - (-1)^n$ is $0$ for even $n$ and $2$ for odd $n$.

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