Fourier series has always been a very key part of infinite series problem, not only can solve many difficult partial differential equations problems, but also has extremely important applications in signal processing, thermodynamic statistical physics, quantum physics and other disciplines. Based on the basic mathematical analysis knowledge such as integral, this paper focuses on the theoretical basis of Fourier series and its application and extension in solving infinite series sums. The main part of the paper will focus on the derivation of mathematical formulas, accompanied by text explanations and the author’s thinking. In this paper, the author proves the convergence theorem and the discriminant method of practical significance, and applies the theoretical knowledge to the concrete calculation of series. At the same time, the author has also found many problems, and from different angles to explore the possible ways to solve the problem of series. These basic calculation ideas and results are instructive to the application of Fourier series in other fields.
