I'm trying to find the fourier series of |sin x| between -pi and pi.(adsbygoogle = window.adsbygoogle || []).push({});

I've got it down to:

[tex]

a_n = \frac{1}{\pi} \int^{\pi}_{-\pi} |sin x| cos (nx) dx

[/tex]

which i wrote as:

[tex]

a_n = \frac{2}{\pi}\int^{\pi}_0 sin x cos (nx) dx

[/tex]

writing

[tex]

sin x cos (nx) = \frac{1}{2} (sin (n+1)x - sin (n-1)x)

[/tex]

I eventually get

[tex]

a_n = \frac{2(n(-1)^n - 1)}{\pi(n^2 - 1)}

[/tex]

giving

[tex]

f(x) = \frac{2}{\pi} + \sum^{\infty}_{n=1} \frac{2(n(-1)^n - 1)}{\pi(n^2 - 1)} cos(nx)

[/tex]

The answer however gives

[tex]

f(x) = \frac{2}{\pi} + \frac{4}{\pi}\sum^{\infty}_{n=1} \frac{cos (2nx)}{4n^2 - 1}

[/tex]

I don't see how they arrive at this... if anyone can let me know where i've gone wrong or if i'm missing something :S

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# Fourier series of |sin x|

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