Calculus II · Unit 4A · lesson
The Direct Comparison Test
Choose comparison inequalities in the useful direction and prove convergence or divergence with familiar benchmark series.
Section overview
Positive-series tests and error estimatesWhat this section is building
Choose comparison inequalities in the useful direction and prove convergence or divergence with familiar benchmark series.
Direct comparison transfers inequalities; limit comparison transfers asymptotic scale; the integral test links sums to accumulated area.
Use a clean inequality when available, asymptotic comparison when ratios stabilize, and the integral test when a matching decreasing function is natural.
Reversing the direction needed to prove convergence or divergence, or forgetting an integral-test remainder condition.
Learning objectives
prove convergence or divergence by bounding a positive series with a known benchmark.
The Direct Comparison Test
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 -series by keeping the highest powers. Square roots and logarithms may require simple bounds valid only for sufficiently large , which is enough because finite initial terms do not affect convergence.
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.
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.
Comparison is really a statement about partial sums
If , then for every partial sum. A convergent bounds the increasing sequence ; a divergent forces upward with it.
Direct Comparison Test
For : if converges, then converges. If and diverges, then diverges.
A convergent rational series
For ,
Because converges, converges by direct comparison.
Build an upper bound for convergence
Test
For , , so
The benchmark converges. Therefore the given series converges by direct comparison.
A smaller divergent series tells you nothing
From and divergence of , no conclusion follows. A convergent series can sit below a divergent one.
u4a-direct_comparison_test-01Classify .
Your work stays on this device. No account or AI grader is used.
Show hint
Compare termwise with .
Attempt once to unlock the solution
Submit an answer first. The hint is available now.
Show that converges.
Show that diverges by comparison with a harmonic multiple.
Explain why cannot prove convergence.
Find a simple comparison for .
Source & rights
Original instruction with traceable references.
BetterGrades-original; no direct adaptation declared in the verified handoff.
Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.