 Fourier series of x cubed minus pi squared x

ID: fourier-series-of-x-cubed-minus-pi-squared-x

Codex  0 Created 2026-09-24 Updated 2026-09-24

(- 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} .

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