Section 2.6 explains the link between the smoothness of a function and the rate of convergence of its Fourier series. This provides a useful check for your Fourier series calculations.



Section 2.7 sums the partial sums of a Fourier series to provide some geometric insight into the cause of Gibb’s phenomenon, which is universal for functions with jump discontinuities and awful for approximation purposes.



Section 2.8 describes the straightforward generalisation of the theory of Fourier series to functions of any period, illustrated by a worked example.



Section 2.9 describes the half-range series that will be used latter on in the course to solve PDE problems. A worked example draws together and illustrates the most important elements of section 2.