The Fourier Series is widely used in math, science, and engineering—some would even say it is one of the most elegant and remarkable pieces of mathematics. I wrote a formal expository paper on Fourier Series targeting students who have taken a real analysis course.

From formally defining the Dirichlet and Fejér kernels to proving the various convergence criterias, I provided motivation for every step along the way. Without losing mathematical rigor, I provided intuitive figures and explanations for challenging concepts like the Riemann-Lebesgue Lemma and the Riemann Localization Theorem. I aimed to make the rigor of mathematical proofs an elegant and essential part of understanding rather than a dry and annoying part about higher-level math.