Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2005 / Paper 1 / 12 / c

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- Solution c

Solution

Evaluate the Fourier cosine series of a triangular wave at its continuous peak

x = 0

, where

f (0) = 1

. Every cosine is one, so

1 = \frac{1}{2} + \frac{4}{π ^{2}} \sum_{m = 0}^{\infty} \frac{1}{( 2 m + 1 ) ^{2}}, m = 0 \sum \infty \frac{1}{( 2 m + 1 ) ^{2}} = \frac{π ^{2}}{8} .

The convergence established above justifies substituting this point into the series.

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