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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. – Stefan Smith Oct 25 '12 at 1:12
up vote 1 down vote accepted

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. – Beat Dec 12 '11 at 4:49
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. – Zarrax Dec 12 '11 at 4:59
Just a remark, in the next to the last sentence you should mention that $|\sin(x)|$ is pwcd and continuous. – AD. Dec 12 '11 at 12:15
Thank You @ Zarrax and AD – Beat Dec 12 '11 at 12:22
@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? – Stefan Smith Apr 6 '13 at 3:01

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