* Expansion into eigenmodes (40 min)

Consider the vector $f \in {\cal H}$, $f(x)= c x (L - x)$.

a) Calculate the constant $c$ such that $f$ is normalized, i.e. $\Vert f\Vert = 1$. Show that $c=\sqrt{{30}/{L}}/{L^2}.$

b) Show that $f$ can be expanded in the basis $\psi_n$ as

$$f=\sum_{n=1}^{\infty} c_n \psi_n, \quad c_n = 2 \sqrt{60} \frac{1-(-1)^n}{n^3\pi^3}$$ (27)

c) Use b) to prove the formula

$$\frac{\pi^3}{32}= \sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^3}.$$