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I need to prove that for every $0 \le x \le \pi$ $$\sum^{\infty}_{n=-\infty}\frac{e^{2inx}}{1-4n^2}=\frac{\pi}{2}\sin{x}$$ using Fourier series.

Let $f(x)$ be a function such that its Fourier series is $$f(x)\sim\sum^{\infty}_{n=-\infty}\frac{e^{2inx}}{1-4n^2}=\sum^{\infty}_{n=-\infty}\frac{1}{1-4n^2}e^{2inx}=\sum^{\infty}_{n=-\infty}\frac{1}{1-4n^2}e^{in\pi x/(\pi/2)}$$ So I know $L=\pi/2$, $b_n=-2Im(\frac{1}{1-4n^2})=0$, and $a_n=2Re(\frac{1}{1-4n^2})=\frac{2}{1-4n^2}$ ($a_n$ and $b_n$ are the Fourier coefficients) hence the the series is of the form $$f(x)\sim \sum^{\infty}_{n=0}\frac{2}{1-4n^2}\cos{2nx}$$

I assume that in order to prove the required convergence I need to find $f(x)$ such that its Fourier series is as described above, but this is a series of an even function(it has only cosine) and $\frac{\pi}{2}\sin{x}$ is clearly an odd function, thus I don't understand how can I find such $f(x)$

CforLinux
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  • $x\mapsto \frac\pi 2 \sin x$ is odd if defined on $(-\pi, \pi)$, but here you're asked to consider $(0,\pi)$. Thus consider the $\pi$-periodic function defined by $\frac \pi 2 \sin x$ on $(0, \pi)$. – Stefan Lafon Jan 03 '22 at 22:10

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For $0\le x\le\pi$, \begin{align} \sin(x)=|\sin(x)|&=\frac2{\pi}-\frac4{\pi}\sum_{n=1}^\infty\frac{\cos(2nx)}{4n^2-1}\\ &=\frac2{\pi}-\frac2{\pi}\sum_{n=1}^\infty\frac{\cos(2nx)+\cos(-2nx)+i\sin(2nx)+i\sin(-2nx)}{4n^2-1}\\ &=\frac2{\pi}-\frac2{\pi}\left(1+\sum_{n=-\infty}^\infty\frac{\cos(2nx)+i\sin(2nx)}{4n^2-1}\right)\\ &=\frac2{\pi}\sum_{n=-\infty}^\infty\frac{\cos(2nx)+i\sin(2nx)}{1-4n^2}\\ \end{align}

Kay K.
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  • But how do you prove the convergence of the Fourier series? $\left| \sin(x) \right|$ doesn't have a $2\pi$ period, so Fourier theorem isn't applicable – CforLinux Jan 04 '22 at 10:12
  • $2\pi$ is one of the periods that $|\sin(x)|$ has. Its 'minimum' period is $\pi$. Whether you regard its period to be $\pi$, $2\pi$, or $n\pi$, you should get the same series. BTW, the Fourier series of the $|\sin(x)|$ is the first example in https://en.wikipedia.org/wiki/Fourier_series – Kay K. Jan 04 '22 at 13:20