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5. Fourier series

a) You are given that for any real numbers $a$ and $b$, \begin{align*} \sin(a+b)x + \sin(a-b)x &= 2\sin ax \cos bx \\ \cos(a+b)x + \cos(a-b)x &= 2\cos ax \cos bx \\ \cos(a-b)x - \cos(a+b)x &= 2\sin ax \sin bx. \end{align*}

The continuous periodic function $f$ with period $2L$ is defined by \[ f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty [a_n \cos(n\pi x/L) + b_n\sin(n\pi x/L)]. \]

Assuming that the orders of summation and integration are interchangable, find $a_n$ and $b_n$.

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b) Sketch the graph of the $2\pi$-periodic function $f$ defined by \[ f(x) = \begin{cases} 0 & -\pi < x < 0 , \cos x & 0 \leq x \leq \pi. \end{cases} \]

Indicate on the graph the values to which the Fourier series of $f$ converges at any point $x$ where it does not converge to $f(x)$. Find the Fourier series of $f$ and hence find the sum \[ \sum_{m=1}^\infty \frac{m \sin (2mx)}{4m^2 - 1}, \quad x \in (0, \pi). \]

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