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

Graphical or geometric meaning

Partial sums of the target stay below corresponding partial sums of the convergent benchmark. A ceiling on every partial sum leads to convergence because positive-term partial sums are increasing and bounded.

Common mistakes and why they fail

Verification and reasonableness checks

• Test the inequality algebraically for the stated range.
• Name the benchmark series and its parameter.
• Ask whether the target is the smaller series for convergence or larger for divergence.

Match inequality direction to the conclusion

For eventually nonnegative terms, prove convergence with a known convergent upper bound $0\le a_n\le b_n$. Prove divergence with a known divergent lower bound $0\le b_n\le a_n$. Reversing either inequality proves nothing: being below a divergent series or above a convergent one leaves both outcomes possible. Finite initial terms do not affect convergence, so an inequality may hold only beyond some index. Verify positivity first; sign-changing series need absolute values or another test. Rational terms often compare with the power determined by leading degrees, but state an actual inequality or use limit comparison rather than relying on resemblance. Finish by naming the benchmark series and why its behavior is known.

Practice

1. Does $\sum1/(n^3+5)$ converge?
2. Does $\sum1/\sqrt{n}$ converge?
3. Can $a_n\le b_n$ and divergence of $b_n$ prove divergence of $a_n$?

Connections and next steps

• Limit Comparison Test
• Integral Test
• Absolute and Conditional Convergence