The Direct Comparison Test

Think of a convergent series as a sum that settles on a finite total. If every term of your series is smaller than or equal to the matching term of that settled sum, your running total can never grow past it either. It's boxed in, so it has nowhere to go but to a finite limit of its own. Flip it around for divergence: if your terms are always at least as big as the terms of a series that blows up, your sum gets dragged up to infinity along with it. The test never tells you what the sum equals, only whether it exists.

To use it, pick a comparison series whose behavior you already know cold, almost always a p-series (sum of 1/n^p) or a geometric series. Look at the formula for large n and ask what it looks like once the small stuff stops mattering. Then prove the inequality: term ≤ comparison term for convergence, or term ≥ comparison term for divergence, for all n past some starting point (the first few terms never matter for convergence). Both series must have nonnegative terms, or the squeeze logic breaks down.

This test sits next to the limit comparison test, which handles cases where the clean inequality is messy to prove directly, and next to the integral test, which compares a series to an integral instead of another series. All three lean on the same instinct: match unfamiliar behavior to familiar behavior. Mastering p-series and geometric series first is what makes this whole family of tests usable.