What Are the Series Comparison Tests? - Expii

The idea behind comparison tests is to compare a complicated series to a simpler one. The direct comparison test says that if \((a_n),(b_n)\) are two sequences of nonnegative numbers such that \(0\le a_n\le b_n\) for all \(n\), then: (1) if the series \(\sum b_n\) converges, then so does the series \(\sum a_n\); or put another way, (2) if the series \(\sum a_n\) diverges, then so does \(\sum b_n\). The limit comparison test says that if \((a_n),(b_n)\) are two sequences of positive numbers such that the limit \(\lim_{n\to\infty} \frac{a_n}{b_n}\) exists and equals a finite nonzero (positive) number, then the two series \(\sum a_n,\sum b_n\) either both converge or both diverge.