10.6 Comparison Tests for Convergence

The direct comparison test compares a series with positive terms to another series whose convergence behavior is already known. If ${0\le a_n \le b_n}$ for all sufficiently large n, and if ∑ b_n converges, then ∑ a_n also converges. On the other hand, if ${0\le b_n \le a_n}$ and ${\sum b_n}$ diverges, then ${\sum a_n}$ also diverges. The idea is simple: a smaller positive series cannot diverge if a bigger one already converges, and a bigger positive series cannot converge if a smaller one already diverges.

To use this test well, you should look for a comparison series that resembles a geometric series or a p-series. The challenge is not the logic of the test, but choosing the right comparison. Usually you compare only the dominant part of the term, especially for rational expressions. Since the test requires positive terms, it is mainly used for nonnegative series.