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I was solving this question I saw in a textbook. The question is :

Calculate the Fourier series for $ f(x) = |\sin x| $ for $-\pi \leq x \leq \pi$.

Which I had $ f(x) = \frac{a_{0}}{2} + \sum a_{n} \cos nx$, where $a_{n} = \frac{2}{\pi} \int_{0}^{\pi} |\sin x|\cos nx dx$. Which I have been able to do; that is by using trig substitution. I had $$ \frac{(n-1)[(-1)^{n+1} - 1] + [(-1)^{n+1} -1)](n+1)}{\pi (n^2 - 1)}$$

For the convergence of $f(x)$, I know it convergences at $x = 0$ because the function is even continuous function.That is by using $$\frac{f(-\pi) + f(\pi)}{2}.$$ Now the problem is, how do I use the Fourier series in above to show that $ \sum_{1}^{\infty} \frac{1}{4n^{2} - 1} = \frac{1}{2}$. I really need guidelines.

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    The series can be evaluated more easily using a telescoping series, in case you are interested.2012-10-25

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Use the fact that $\sin(x)$ is nonnegative on $0 \leq x \leq \pi$, so that your $a_n$ is given by $$a_n = {2 \over \pi}\int_0^{\pi}\sin(x)\cos(nx)\,dx$$ For $n=0$, compute this directly. Otherwise use $\sin(a)\cos(b) = {\sin(a + b) + \sin(a-b) \over 2}$. The Fourier series converges to $|\sin(x)|$ everywhere because it's piecewise continuously differentiable. Plug in $x = 0$ into the Fourier series to get the summation.

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    Yes, @ Zarrax, I understand that part, the problem is , how do I use the Fourier series to show that the summation is $ \frac{1}{2}$. $a_{0} = \int_{0}^{\pi} sinx dx$ which I can do.2011-12-12
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    Plug in $x = 0$ into your cosine series. You get $|\sin(0)| = {a_0 \over 2} + \sum_n a_n$. Use that $\sin(0) = 0$ and the summation here will be related to the series you're trying to add up.2011-12-12
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    Just a remark, in the next to the last sentence you should mention that $|\sin(x)|$ is pwcd and *continuous*.2011-12-12
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    @Zarrax : Do you really need piecewise continuously differentiable or is continuous, or uniformly continuous, good enough to get pointwise convergence of the Fourier series to $f(x)$? I often teach this material to engineering students. Should they care about the pointwise convergence (I don't) or just about the $L^2$ convergence?2013-04-06
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    There are examples of uniformly continuous functions whose Fourier series diverge on a dense set. If you have Katznelson's book he describes this in the remark after the proof of Theorem 2.1. More generally, pointwise convergence behavior can be relevant due to such things as Gibbs' Phenomenon.2013-04-07