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CalculusBy Educator Today

C44 · Unit C10 Sequences and series

Direct and limit comparison

BC extensionReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 10.6 · BC
Builds on
Part of

Progress on this lesson counts in every pathway listed.

Understand the idea

For nonnegative terms, if 0 is less than or equal to a sub n is less than or equal to b sub n and sum b sub n converges, then sum a sub n converges. If a sub n is greater than or equal to b sub n is greater than or equal to 0 and sum b sub n diverges, then sum a sub n diverges. The opposite inequality directions do not decide the question. Limit comparison with positive b sub n works when a sub n /b sub n tends to L with 0<L<∞; then the series have the same convergence behavior.

Worked example

For a sub n equals (2n+1)/(n cubed +4), compare with 1 over n squared. Their ratio is (2n cubed +n squared)/(n cubed +4) tends to 2, so the series converges. The nonzero finite limit captures the dominant power without claiming the terms are identical.

Explore the mathematics

Direct and limit comparison

Compare positive terms to a known convergent benchmark.

Setting 3 of 5

Scroll sideways to see the full diagram.

How can a known series help classify another positive-term series?12345Blue: 1/(k²+1); yellow: 1/k²

Predict before continuing

How can a known series help classify another positive-term series?

Read the complete explanation
  1. How can a known series help classify another positive-term series?
  2. A direct comparison requires a valid inequality in the correct direction.
  3. For limit comparison, a finite positive limit of the term ratio links the two convergence behaviours.
  4. For k≥1, 0<1/(k²+1)<1/k². The convergent p-series with p=2 bounds the blue series from above, proving convergence.
See the idea: Direct and limit comparison
See the idea: Direct and limit comparisonThe smooth guide plots the ratio aₙ/(1/n²)=(2n³+n²)/(n³+4); the sequence uses integer n. It tends to the positive finite value 2, so limit comparison with the convergent p-series applies.xy103.750.6256.51.259.251.875122.5

The smooth guide plots the ratio aₙ/(1/n²)=(2n³+n²)/(n³+4); the sequence uses integer n. It tends to the positive finite value 2, so limit comparison with the convergent p-series applies.

Discuss: explain what each label represents, then draw a related example with different values.

A common misconception

Being smaller than a divergent series proves nothing. Being larger than a convergent series also proves nothing.

Investigate and explain

For each comparison argument, label which series is known and which inequality direction supports the claim.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.