Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 2 / 7

Table of contents
- a 7
  - Solution a
- b 7
  - Solution b

a

Solution

Use the double-angle identity

2 sin x cos x = sin 2 x

. The Fourier series is already a finite trigonometric sum:

f (x) = sin x + sin 2 x + sin 3 x .

In the usual Fourier sine series convention,

b_{1} = b_{2} = b_{3} = 1

, all other

b_{n}

vanish, and all cosine coefficients and the constant coefficient vanish.

b

Solution

Differentiation of the finite sum involves no convergence issue. The derivative has the Fourier cosine series

f^{'} (x) = cos x + 2 cos 2 x + 3 cos 3 x .

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