The D'Alembert Criterion

This lecture introduces the D'Alembert criterion, which is used to determine the convergence or divergence of a series based on the limit of the ratio of consecutive terms. The criterion states that if the limit is less than 1, the series converges absolutely, and if it is greater than 1, the series diverges. The proof of the criterion involves establishing conditions for convergence and divergence based on the limit value. Additionally, a remark is made regarding the case where the limit equals 1, indicating different convergence outcomes based on the behavior of the terms.