Sum of reciprocals of fourth powers

Calculus Level 2

Find the exact sum of $\sum_{n=1}^\infty \frac{1}{n^4}= 1+\frac{1}{2^4} + \frac{1}{3^4} + \frac{1}{4^4} + \cdots$ also known as $\zeta(4)$ , where $\zeta(s)$ is the Riemann zeta function.

Hint: Follow Euler's famous solution of the Basel problem.