A convergence test is a method for deciding whether an infinite series converges (adds to a finite value) or diverges, based on how its terms behave. The AP Calculus BC exam assesses six: the nth term test for divergence, integral test, comparison test, limit comparison test, alternating series test, and ratio test.
A convergence test answers one question about an infinite series: does adding up infinitely many terms produce a finite number, or does the sum blow up (or bounce around forever)? You can't actually add infinitely many terms by hand, so each test gives you a shortcut. It looks at the pattern of the terms and tells you the series' fate without computing the sum.
The AP Calculus BC exam (this is BC-only material, Unit 10) assesses exactly six tests: the nth term test for divergence, the integral test, the comparison test, the limit comparison test, the alternating series test, and the ratio test (LIM-7.A.11). The CED explicitly excludes everything else, so tests like the root test won't appear on the exam. Most of the skill here isn't running a test; it's picking the right one. Factorials and exponentials scream ratio test. A series that looks like 1/n^p is a p-series. Terms that alternate signs point to the alternating series test.
Convergence tests live in Unit 10: Infinite Sequences and Series, and they directly support learning objective 10.8.A, "Determine whether a series converges or diverges." Topic 10.8 focuses on the ratio test specifically, but the broader toolkit spans Topics 10.3 through 10.9. This material is the gatekeeper for the back half of Unit 10. You can't find the interval of convergence of a power series or trust a Taylor series approximation until you can decide whether a series converges in the first place. Unit 10 carries serious weight on the BC exam, and convergence questions are some of the most reliable multiple-choice points if you know which test to reach for.
Ratio Test (Unit 10)
The ratio test is the convergence test the CED gives its own topic (10.8), because it's the workhorse. You take the limit of |a(n+1)/a(n)|; less than 1 means converges, greater than 1 means diverges, equal to 1 means the test tells you nothing. It later becomes the standard tool for finding a power series' radius of convergence.
Improper Integrals (Unit 6)
The integral test is the bridge between Units 6 and 10. If f(x) is positive, continuous, and decreasing, the series Σf(n) and the improper integral of f(x) from 1 to infinity share the same fate. Series convergence is secretly an improper integral problem in disguise.
Limits (Unit 1)
Every convergence test is built on a limit. The nth term test checks the limit of the terms, the ratio test takes a limit of consecutive-term ratios, and the limit comparison test is a limit by name. Unit 10 is where your Unit 1 limit skills come back for a final exam of their own.
Radius and Interval of Convergence (Unit 10)
Convergence tests aren't just for numerical series. Apply the ratio test to a power series and you get the radius of convergence, then check the endpoints with other tests (often the alternating series test or p-series facts). One skill, recycled twice.
Convergence tests show up heavily in BC multiple choice, usually in two flavors. One flavor hands you a series and asks whether it converges or diverges, so you have to choose and execute the right test. The other flavor tests whether you know the tests themselves, with stems like "which test should you use for a series of the form 1/n^p?" or "what does the ratio test state?" On the FRQ side, convergence reasoning typically appears inside Taylor/power series problems, where you justify an interval of convergence using the ratio test and endpoint checks. Justification matters. Saying "it converges" earns nothing; you have to name the test, verify its conditions, show the limit, and state the conclusion. And remember the exclusion statement: only the six listed tests are fair game, so you'll never be forced to use the root test.
The nth term test is a one-way street, and treating it as two-way is the most common convergence error on the exam. If the terms don't approach 0, the series diverges, full stop. But if the terms DO approach 0, you know nothing yet. The harmonic series Σ1/n is the classic trap: its terms go to 0, and it still diverges. The nth term test can only ever prove divergence, never convergence.
A convergence test determines whether an infinite series converges or diverges by analyzing the behavior of its terms, without computing the actual sum.
The BC exam assesses exactly six tests: nth term test for divergence, integral test, comparison test, limit comparison test, alternating series test, and ratio test. Nothing else is tested.
Match the test to the series' structure: ratio test for factorials and exponentials, p-series rules for 1/n^p, alternating series test for terms that switch signs.
The nth term test only proves divergence. Terms going to 0 does NOT prove convergence, and the harmonic series is the counterexample to memorize.
When the ratio test's limit equals 1, the test is inconclusive and you must switch to a different test.
Convergence tests power the rest of Unit 10, since you need them to find the interval of convergence of every power series and Taylor series.
It's a method for deciding whether an infinite series adds up to a finite value (converges) or not (diverges) by examining its terms. The BC exam assesses six specific tests under learning objective 10.8.A and the surrounding Unit 10 topics.
Exactly six: the nth term test for divergence, the integral test, the comparison test, the limit comparison test, the alternating series test, and the ratio test. The CED's exclusion statement says other methods, like the root test, are not assessed.
No. Terms approaching 0 is necessary for convergence but never sufficient. The harmonic series Σ1/n has terms that go to 0 and still diverges, which is exactly the trap the exam loves to set.
A convergence test only tells you whether a finite sum exists, not what it equals. On the BC exam you only compute actual sums in special cases, like geometric series or recognizable Taylor series values.
No. All of Unit 10, including every convergence test, is BC-only material. If you're taking AB, you'll never see a series convergence question.
Treat it as a p-series: it converges when p > 1 and diverges when p ≤ 1, a fact you can justify with the integral test. Don't reach for the ratio test here, since it comes out inconclusive (the limit equals 1).