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

The periodic extension is an even triangular wave: peaks of height one occur at the even integers $x=2k$ and valleys of height zero at the odd integers $x=2k+1$. Adjacent peaks and valleys are joined by straight segments of slopes $\pm 1$. The function is continuous and has corners at every integer.

Evenness removes every sine coefficient. In the period-two Fourier cosine series convention $f=a_0/2+{\sum }_{n\ge 1}{a}_n\cos (n\pi x)$, the constant coefficient is $a_0=2{\int }_0^1(1-x)dx=1$. Integration by parts givesThus the Fourier cosine series of a triangular wave isThe periodic function is continuous and piecewise smooth, so the Dirichlet-Jordan convergence theorem gives its value at every point. The summable absolute coefficient bound also gives uniform convergence by the Weierstrass M-test.

Evaluate the Fourier cosine series of a triangular wave at its continuous peak $x=0$, where $f(0)=1$. Every cosine is one, soThe convergence established above justifies substituting this point into the series.