Methods | Part IB, 2001

The even function f(x)f(x) has the Fourier cosine series

f(x)=12a0+n=1ancosnxf(x)=\frac{1}{2} a_{0}+\sum_{n=1}^{\infty} a_{n} \cos n x

in the interval πxπ-\pi \leqslant x \leqslant \pi. Show that

1πππ(f(x))2dx=12a02+n=1an2\frac{1}{\pi} \int_{-\pi}^{\pi}(f(x))^{2} d x=\frac{1}{2} a_{0}^{2}+\sum_{n=1}^{\infty} a_{n}^{2}

Find the Fourier cosine series of x2x^{2} in the same interval, and show that

n=11n4=π490\sum_{n=1}^{\infty} \frac{1}{n^{4}}=\frac{\pi^{4}}{90}

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