Fourier series of x cubed minus pi squared x

(- π, π)

x^{3} - π^{2} x = 12 \sum_{n = 1}^{\infty} \frac{( - 1 ) ^{n}}{n ^{3}} sin (n x) .

Parseval's identity and direct integration of the square give

\sum_{n = 1}^{\infty} \frac{1}{n ^{6}} = \frac{π ^{6}}{945} .

 Ancestors (5)