Consider the function \(f(x)=x, 0

a. Find the Fourier sine series representation of this function and plot the first 50 terms.

b. Find the Fourier cosine series representation of this function and plot the first 50 terms.

c. Extend and apply Parseval's identity in Problem 8 to the result in part \(b\).

d. Use the result of part

c, to find the sum \(\sum_{n=1}^{\infty} \frac{1}{n^{4}}\).

Data from Problem 8

Let \(f(x)\) be defined for \(x \in[-L, L]\). Parseval's identity is given by

\[\frac{1}{L} \int_{-L}^{L} f^{2}(x) d x=\frac{a_{0}^{2}}{2}+\sum_{n=1}^{\infty} a_{n}^{2}+b_{n}^{2}\]

Assuming the the Fourier series of \(f(x)\) converges uniformly in \((-L, L)\), prove Parseval's identity by multiplying the Fourier series representation by \(f(x)\) and integrating from \(x=-L\) to \(x=L\). [We will encounter Parseval's equality for Fourier transforms, which is a continuous version of this identity.]