Please show all of your work !!!! We shall find a Fourier series for the function f(x) = x on the interval x \in (?2, 2) extended to a periodic function:

Please show all of your work !!!!

We shall find a Fourier series for the function f(x) = x on the interval x \in (?2, 2) extended to a periodic function:

f(x) = -\frac{4}{\pi }\sum_{n=1}^{\infty }\frac{1}{n}(-1)^{n}sin\left ( \frac{n\pi x}{2} \right ) (1)

One application, perhaps unexpected, of such a Fourier series is the evaluation of infinite sums, for instance

S = 1 - \frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9} (2)

resembling another conditionally-convergent series in the context of the binding energy of crystalline salt. Evaluate (2) by substituting a particular value for x in (1). (We

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