The Limit Comparison Test (BC only, Topic 10.6) determines whether a series with positive terms converges or diverges by computing the limit of aₙ/bₙ against a known series. If that limit is positive and finite, both series converge or both diverge together.
The Limit Comparison Test (LCT) is a convergence test for infinite series with positive terms. You take the series you're stuck on, Σaₙ, pick a comparison series Σbₙ whose behavior you already know (usually a p-series or geometric series), and compute the limit of aₙ/bₙ as n goes to infinity. If that limit is a positive, finite number, the two series do the same thing. Both converge or both diverge.
Here's the intuition. If aₙ/bₙ settles on some constant like 3, then for large n the terms aₙ are basically just 3 times bₙ. Multiplying a series by a constant doesn't change whether it converges, so the two series share a fate. That's why the LCT is the go-to move for messy rational-style terms like (2n + 1)/(n³ + 5n). You strip away the junk, compare to 1/n², and you're done. The CED lists it under essential knowledge LIM-7.A.9 as one of the core methods for determining convergence.
The Limit Comparison Test lives in Topic 10.6 (Comparison Tests for Convergence) in Unit 10: Infinite Sequences and Series, which is BC-only. It directly supports learning objective 10.6.A ("Determine whether a series converges or diverges") through essential knowledge LIM-7.A.9. Unit 10 is the single biggest chunk of the BC exam by weight, and convergence testing is its backbone. The LCT matters because it rescues you when the direct comparison test fails on a technicality. If your terms are slightly bigger than a convergent series or slightly smaller than a divergent one, direct comparison tells you nothing, but the LCT usually still works. It also shows up downstream whenever you test endpoints of an interval of convergence for a power series.
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Direct Comparison Test (Unit 10)
These two share Topic 10.6 and EK LIM-7.A.8/LIM-7.A.9. The direct comparison test needs a term-by-term inequality to hold; the LCT only needs the ratio of terms to settle at a positive finite number. Think of the LCT as the forgiving cousin you call when the inequality points the wrong way.
Convergent and Divergent Series (Unit 10)
The LCT is useless without a stockpile of known series to compare against. P-series (converge when p > 1) and geometric series (converge when |r| < 1) are your benchmark library. The whole strategy is matching an ugly series to a clean one whose verdict you've memorized.
Ratio Test (Unit 10)
Both tests involve a limit of a ratio, but they're different ratios. The ratio test compares a series to itself (aₙ₊₁/aₙ), while the LCT compares two different series (aₙ/bₙ). Use the ratio test for factorials and exponentials; use the LCT for algebraic terms where the ratio test gives an unhelpful limit of 1.
Taylor Series (Unit 10)
When you find a power series' interval of convergence, the ratio test handles the interior but always fails at the endpoints. Plugging in an endpoint gives you a plain numerical series, and the LCT is often the test that settles whether that endpoint gets included.
The Limit Comparison Test is BC-only and appears mostly in multiple-choice questions on convergence. Expect stems that test the conditions of the test itself, like asking what you can conclude when the limit of aₙ/bₙ is positive and finite (both series behave the same), or what's required before you can even apply the test (positive terms). Watch for edge-case questions too. If the limit of aₙ/bₙ equals infinity or zero, the standard "same behavior" conclusion doesn't apply, and questions love to check whether you know that. No released FRQ has named the test verbatim, but free-response questions on series convergence and intervals of convergence expect you to choose an appropriate test and justify it, and the LCT is frequently the cleanest choice for rational-looking terms. Always state the comparison series, show the limit computation, and name the conclusion.
The direct comparison test requires an actual inequality between terms (aₙ ≤ bₙ or aₙ ≥ bₙ) that holds for all large n, and it only gives a conclusion when the inequality points the useful direction. The Limit Comparison Test skips the inequality entirely and just asks whether aₙ/bₙ approaches a positive finite number. Practical rule. If you can see the inequality instantly, use direct comparison; if the terms have extra pieces like "+1" or "−3" that wreck the inequality, use the LCT.
If the limit of aₙ/bₙ as n approaches infinity is positive and finite, then Σaₙ and Σbₙ either both converge or both diverge.
Both series need positive terms before you can apply the Limit Comparison Test at all.
Choose your comparison series by keeping only the dominant terms, so (3n + 2)/(n³ + n) compares to 1/n², a convergent p-series.
If the limit comes out to 0 or infinity, the standard conclusion doesn't hold, and exam questions specifically test whether you know that.
The LCT works when direct comparison fails on a technicality, like when your terms are slightly larger than a convergent series.
This is BC-only material from Topic 10.6, supporting learning objective 10.6.A through essential knowledge LIM-7.A.9.
It's a convergence test from Topic 10.6 where you compute the limit of aₙ/bₙ between your series and a known series like a p-series. If the limit is positive and finite, both series converge or both diverge together.
Not in the standard form tested on the exam. The "both behave the same" conclusion requires a positive, finite limit. If the limit of aₙ/bₙ equals infinity or zero, you can't draw the usual conclusion, and MCQs love to test this edge case.
Direct comparison needs a term-by-term inequality like aₙ ≤ bₙ to hold, and it fails if the inequality points the wrong way. The LCT only needs the limit of aₙ/bₙ to be positive and finite, which makes it the better tool for terms with pesky added constants.
No. It's part of Unit 10 (Infinite Sequences and Series), which is BC-only content. AB never tests series convergence.
Keep only the dominant terms of the numerator and denominator. For something like (2n + 1)/(n³ + 5n), the dominant behavior is n/n³ = 1/n², so compare to the convergent p-series Σ1/n². P-series and geometric series are your standard benchmarks.