Consider : $\displaystyle f(x)= \sum_{n=1}^{\infty} \frac{\sin nx }{n^4}$
Find : $\displaystyle \int_0^{x} f(t)\ \mathrm{d}t$.
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1Can you find $\int_0^x{\sin nt\over n^4},dt$? – Gerry Myerson Feb 08 '13 at 11:28
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1If $f_N(t)=\sum_{n=1}^N \sin(nt)/n^4$, then $f_N:[0,x]\to\mathbb{R}$ is converges uniformly to $f$ for all $x$. So you can integrate this series term-by-term. – Hanul Jeon Feb 08 '13 at 11:52
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@GerryMyerson : yes I can but how to calculate this : $\displaystyle \sum \frac{\cos {nx} }{n^5} $ – Tulip Feb 08 '13 at 11:55
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@aziiri - if only it were an odd power instead of a power of 4.. – Nathaniel Bubis Feb 08 '13 at 11:58
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Good question. Fourier series? – Gerry Myerson Feb 08 '13 at 12:30
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I've tried Fourier series before and failed : do you know any even function such as : $\displaystyle \int_{-\pi}^{\pi} f(t)\cos nt \ \mathrm{d}t=\frac{\pi}{n^5} $ or just an odd function satisfying the first sum ? – Tulip Feb 08 '13 at 12:34
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@GerryMyerson - for a Fourier series it would seem that you would need an odd power in the denominator of the sine. – Nathaniel Bubis Feb 08 '13 at 12:34
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Note that, $$ \int_{0}^{x}f(t)\ \mathrm dt=\sum_{n=1}^{\infty}\frac{1}{n^4}\int_{0}^{x}\sin(nt) \ \mathrm dt=\sum_{n=1}^{\infty}\frac{1}{n^4}\left( \frac{1}{n}-\frac{\cos(nx)}{n} \right) $$
$$ \implies \int_{0}^{x}f(t) \ \mathrm dt=\sum_{n=1}^{\infty}\frac{1}{n^5}-\sum_{n=1}^{\infty}\frac{\cos(nx)}{n^5} $$
$$\implies f(x)=\zeta(5)-\frac{1}{2}\sum_{n=1}^{\infty} \frac{e^{inx} }{n^5} - \frac{1}{2}\sum_{n=1}^{\infty} \frac{e^{-inx} }{n^5} $$
$$ \implies f(x)=\zeta(5)-\frac{1}{2}( \operatorname{Li}_{5}(e^{ix})+ \operatorname{Li}_{5}(e^{-ix}) ),$$
where $\operatorname{Li}_{s}(z)$ is the Polylogarithm function.
Ali Caglayan
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Mhenni Benghorbal
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this was my first thought, but I assumed that there is a solution without polylogarithms. – Tulip Feb 08 '13 at 14:42
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@aziiri: I do not think you can get it in terms of simpler functions. – Mhenni Benghorbal Feb 08 '13 at 14:51
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@aziiri: By the way, since it was your first thought, why you did not mention it, so we know what we should post. – Mhenni Benghorbal Feb 08 '13 at 15:28
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I've written what I've done but someone said it is useless, so I deleted it. sorry for inconvenience. – Tulip Feb 08 '13 at 16:06
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