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Year 1 of the natural sciences course of the University of Cambridge

Mathematics course of the year 1 of the natural sciences course of the University of Cambridge

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge

2017

Paper 2

For this odd <function>, the <Parseval identity> from part (a) gives
$$
\frac1\pi\int_{-\pi}^{\pi}(x^3-\pi^2x)^2\,dx
=144\sum_{n=1}^\infty\frac1{n^6}.
$$
The integrand is even. Expanding and integrating it gives
$$
\begin{aligned}
\frac2\pi\int_0^\pi(x^6-2\pi^2x^4+\pi^4x^2)\,dx
&=2\pi^6\left(\frac17-\frac25+\frac13\right)\\
&=\frac{16\pi^6}{105}.
\end{aligned}
$$
Dividing by $144$ therefore proves
$$
\boxed{\sum_{n=1}^\infty\frac1{n^6}=\frac{\pi^6}{945}}.
$$
This evaluates the <Riemann zeta function> at six by using the norm of a simple <polynomial>.

Solution

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2017 / Paper 2 / 18 / c / Solution

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2017 / Paper 2 / 18 / c / Solution

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