Comparison Test

The central idea

For nonnegative terms eventually satisfying $0\le a_n\le b_n$, convergence of $\sum b_n$ implies convergence of $\sum a_n$. If $0\le b_n\le a_n$ and $\sum b_n$ diverges, then $\sum a_n$ diverges. The inequalities must point in the direction that transfers the known conclusion.

Definitions, hypotheses, and notation

Finite initial exceptions do not matter; an eventual inequality is enough because convergence depends on the tail. For rational-looking terms, discarding positive denominator pieces makes a fraction larger, while discarding numerator pieces makes it smaller. Tracking that direction helps construct rigorous bounds.

If no convenient global inequality appears, limit comparison may be better. Direct comparison is stronger pedagogically when the inequality is simple because it shows explicit control, but neither test computes the sum; both decide only convergence or divergence.

Conceptual meaning

A smaller positive total cannot exceed a known finite total, while a larger positive total cannot remain finite if a smaller one already grows without bound. Comparison uses magnitude and therefore applies directly to nonnegative series.

A dependable method and decision rule

1. Verify terms are nonnegative from some index onward.
2. Simplify dominant factors to choose a benchmark $p$-series or geometric series.
3. Prove the needed inequality rather than relying on visual similarity.
4. Check that the benchmark's convergence behavior is known.
5. State how the inequality direction transfers that behavior.

Fully worked example

Common mistakes and why they fail