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Manual Derivation of Fourier Series for an Intermediate Value
For an intermediate value $x$ in $[-R,R]$
$$\sum_{k=1}^{\infty}\frac{1}{k^2}=\frac{\pi^2}{6}$$
using manual derivation.
I tried to calculate
$$S(\frac{x}{2})=\int_{ -R}^{R}\cos{kx}\,\mathrm{d}x=\frac{1}{k}\sin{kR}-\frac{1}{k}\sin{kx} $$
$$S(\frac{x}{2})+S(\frac{x}{2})=\frac{\sin{kR}+\sin{kx}}{k^2}-\frac{\sin{kR}+\sin{kx}}{k^2}+\frac{\sin{kR}+\sin{kx}}{k^2}=0$$
Then I have problem with summation of this series
and I’m stuck.
I don’t know how to use sum of cosine function for series in general case.
Any help is appreciated.
A:
You are on the right way:
$$
\begin{align}
S(\frac{x}{2})&+S(\
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