Sum of reciprocals of fourth powers

Calculus Level 3

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.

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

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