Lecture 3: Introduction to Series

This lecture opens a new section of the mathematical analysis course dedicated to the theory of series. Students will be introduced to the fundamental concepts needed to understand how infinite sums can lead to finite results. The first part thoroughly examines numerical sequences and their limits—the foundation for the further study of series. Simple examples are used to explain how to determine whether a sequence converges or diverges. Next, the concept of an infinite series is introduced through the sequence of partial sums. Special attention is given to geometric series—a crucial class of series for which a simple sum formula exists. Telescoping series, where most terms cancel each other out, are also discussed. The lecture covers the divergence test and teaches students to avoid the common mistake of assuming that a term tending to zero guarantees the convergence of the series. The material is illustrated with examples from physics and the theory of infinite decimals.