Introduction
Series play an essential role in mathematical analysis and its applications, providing a powerful tool for approximating functions, solving differential equations, and modeling physical phenomena. In this chapter, we begin by studying numerical series and the methods used to determine convergence or divergence. We then extend these ideas to sequences and series of functions, where the distinction between pointwise and uniform convergence becomes important. Next, we examine power series, which allow many functions to be represented as infinite polynomials and are fundamental in calculus and mathematical physics. Finally, we introduce Fourier series, which express periodic functions as sums of sines and cosines and form the basis of harmonic analysis and signal processing.
Chapter Objectives
At the end of this chapter, students should be able to:
1. Understand the concepts of sequences and numerical series, and determine their convergence or divergence.
2. Apply common convergence tests, such as the comparison test, ratio test, root test, and integral test.
3. Analyze sequences and series of functions, and distinguish between pointwise and uniform convergence.
4. Work with power series, determine intervals of convergence, and represent functions as power series.
5. Understand the basic principles of Fourier series and express periodic functions as trigonometric series.
6. Apply series expansions to solve problems in calculus, physics, and engineering.
