Introduction to Fourier series

Abstract

We start the book by considering the series $\mathop {\Sigma }\nolimits _{n=1}^\infty {\mathrm{sin}(n x) \over n},$ a nice example of a Fourier series. This series converges for all real numbers x, but the issue of convergence is delicate. We introduce summation by parts as a tool for handling some conditionally convergent series of this sort. After verifying convergence, but before finding the limit, we pause to introduce and discuss several elementary differential equations. This material also leads to Fourier series. We include the exponentiation of matrices here. The reader will observe these diverse topics begin being woven into a coherent whole