Calculus II · Unit 4A · lesson

The Direct Comparison Test

Concept

Learning objectives

prove convergence or divergence by bounding a positive series with a known benchmark.

The Direct Comparison Test

Explanation

Compare in the direction that actually proves something

For positive terms, a smaller series than a known convergent series must converge, while a larger series than a known divergent series must diverge. The reverse directions prove nothing. A large series can still converge, and a small series can still diverge. Most comparison errors are direction errors, not algebra errors.

The art is choosing a benchmark that captures the dominant behavior without demanding an unnecessarily sharp inequality. Rational expressions are often compared with $p$ -series by keeping the highest powers. Square roots and logarithms may require simple bounds valid only for sufficiently large $n$ , which is enough because finite initial terms do not affect convergence.

Decision

Comparison direction check

To prove convergence, place the unknown positive terms below a known convergent series. To prove divergence, place them above a known divergent series. An inequality pointing the other way may be true but useless for the conclusion you need.

Bridge

Compare in the direction that answers the question

For nonnegative terms, a smaller series than a convergent benchmark must also converge, while a larger series than a divergent benchmark must also diverge. The inequality direction matters because comparison controls accumulated size, not merely the appearance of the formulas.

A good benchmark captures the dominant structure and is simple enough to classify immediately. Before writing an inequality, decide whether you are trying to prove convergence or divergence. That decision tells you whether you need an upper or lower bound.

Proof idea

Comparison is really a statement about partial sums

If $0\le a_n\le b_n$ , then $0\le A_N\le B_N$ for every partial sum. A convergent $B_N$ bounds the increasing sequence $A_N$ ; a divergent $A_N$ forces $B_N$ upward with it.

Concept

Direct Comparison Test

For $0\le a_n\le b_n$ : if $\sum b_n$ converges, then $\sum a_n$ converges. If $0\le b_n\le a_n$ and $\sum b_n$ diverges, then $\sum a_n$ diverges.

Guided walkthrough

A convergent rational series

For $n\ge1$ ,

0<\frac1{n^2+1}<\frac1{n^2}.

Because $\sum1/n^2$ converges, $\sum1/(n^2+1)$ converges by direct comparison.

Worked example

Build an upper bound for convergence

Test

\sum_{n=1}^{\infty}\frac{1}{n^2+5n}.

For $n\ge1$ , $n^2+5n\ge n^2$ , so

0\le\frac{1}{n^2+5n}\le\frac1{n^2}.

The benchmark $\sum1/n^2$ converges. Therefore the given series converges by direct comparison.

Common mistake

A smaller divergent series tells you nothing

From $0\le a_n\le b_n$ and divergence of $\sum b_n$ , no conclusion follows. A convergent series can sit below a divergent one.

Interactive checku4a-direct_comparison_test-01

Classify $\sum_{n=1}^{\infty}1/(n^2+1)$ .

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Show hint

Compare termwise with $1/n^2$ .

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Exercise

Show that $\sum1/(n^2+n)$ converges.

Exercise

Show that $\sum1/\sqrt{n^2+1}$ diverges by comparison with a harmonic multiple.

Exercise

Explain why $a_n\le1/n$ cannot prove convergence.

Exercise

Find a simple comparison for $(3n+2)/(n^3+1)$ .

After the explanation

Use the section idea

Reading lens

Compare positive terms by long-run size and select a benchmark whose convergence behavior is already known.

Mental model

Direct comparison transfers inequalities; limit comparison transfers asymptotic scale; the integral test links sums to accumulated area.

Decision

Use a clean inequality when available, asymptotic comparison when ratios stabilize, and the integral test when a matching decreasing function is natural.

Common trap

Reversing the direction needed to prove convergence or divergence, or forgetting an integral-test remainder condition.

Check yourself

Does your benchmark support the conclusion in the direction you claim, and are all hypotheses stated?

Source & rights

Original instruction with traceable references.

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Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.