Lecture 4: Convergence Tests for Series

This lecture presents the fundamental convergence tests for infinite series, providing students with systematic tools to determine convergence or divergence. The material is intended for students studying mathematical analysis who are already familiar with basic series concepts.

The lecture introduces four essential tests. The Ratio Test handles series with factorials and exponentials, while the Root Test addresses terms raised to powers of n. The Integral Test connects series to improper integrals, enabling analysis of p-series. The Alternating Series Test covers series with alternating signs, including the alternating harmonic series.

Each test is rigorously stated and accompanied by worked examples demonstrating its application. Special attention is given to inconclusive cases. The lecture concludes with guidance on selecting the appropriate test and a summary table for quick reference.