Evaluate ∑n=1∞1n4 using Parseval's theorem (Fourier series).
Evaluate using Parseval's theorem (Fourier series).
Answer & Explanation
Let for . The results of computing the Fourier coefficients
for , and
Therefore for and
We have then
Here and are Fourier coefficients of f.
Let for . Then and
The series also appears in the following funny way. Consider the operator on s.t. Dirichlet boundary conditions. Find Green's function . Find the normalized in eigen functions of the operator; the eigenvalues are .Evaluate the integral in two ways, a) using the explicit formula for and b) using the representation .After simplifications, you will find .