AP Calculus BC Series Tests: Which Convergence Test to Use (Decision Guide)

Quick answer: For AP Calculus BC, always start with the nth-term test (if the terms don't go to 0, the series diverges), then match the series to a shape: geometric (a times r^n), p-series (1/n^p), factorials or n-th powers (ratio test), alternating signs (alternating series test), or a messy rational expression (limit comparison test). The real exam trap isn't picking the right test — it's forgetting to name it and check its conditions, which is where most of the marks actually live.

When I first hit the series unit, it felt like a magic trick everyone else knew the secret to. Eight tests, all looking similar, and the answer to "which one do I use?" was always an infuriating "it depends." What finally clicked for me: you don't choose a test by trial and error — you read the shape of the terms, and the test more or less picks itself.

This is the decision flow I actually use, plus the one thing that quietly costs BC students the most marks: the justification.

Step 0: always run the nth-term test first

The nth-term test (also called the test for divergence) is your free first move. Look at the general term a_n and take its limit:

• If lim (n to infinity) a_n is not 0 → the series diverges. Done, stop here.
• If lim a_n = 0 → the test is inconclusive. You've learned nothing; move on.

The trap that catches everyone at least once: a_n going to 0 does NOT prove convergence. The harmonic series sum 1/n has terms going to 0 and still diverges. So the nth-term test can only ever kill a series, never save it. Use it to catch the easy divergences, then reach for the real tools.

Read the shape, then pick the test

Here's the actual flow. Scan the general term for these patterns, roughly in this order:

• A constant ratio between terms, a times r^n: geometric series. Converges if and only if |r| < 1, and then it sums to a/(1 - r). This is the only common test that also hands you the exact sum.
• 1/n^p (a power of n on the bottom): p-series. Converges if and only if p > 1. So 1/n^2 converges, 1/n (the harmonic series) diverges, and 1/sqrt(n) = 1/n^(1/2) diverges because p = 1/2.
• Factorials (n!) or something raised to the n-th power (2^n, 3^n, n^n): ratio test. Factorials cancel beautifully inside a ratio, so this is almost always the move when you see one.
• An alternating sign, (-1)^n or (-1)^(n+1): alternating series test. Then separately ask whether it converges absolutely (test the series of |a_n|) or only conditionally.
• A rational or algebraic mess (polynomial over polynomial, roots, and so on): limit comparison test against the p-series you get by keeping only the highest powers. Use direct comparison instead if the inequality is clean and obvious.
• a_n = f(n) where f is positive, continuous, decreasing and easy to integrate (like 1/(n ln n)): integral test.

The shape-to-test mapping is worth drilling until it's automatic — I turned mine into Flashcards and tested myself until I could name the right test in under two seconds.

The tests students misuse

Two honest warnings from my own revision. First, direct comparison is fiddly because the inequality has to point the right way: a larger convergent series proves convergence, a smaller divergent one proves divergence, and the other two combinations tell you nothing. When in doubt, limit comparison is far more forgiving.

Second, and this surprises people: the root test is not on the AP Calculus BC exam. It's in plenty of textbooks and it's handy for terms like (something)^n, but it isn't in the College Board course framework, so anything it could do on the exam, the ratio test does too. Don't burn revision time on a test they aren't allowed to ask you about.

The mark BC students throw away: justification

Here's what nobody tells you until you've already lost the points. On the free-response section, "it converges" is usually worth just one point. The rest are for the justification, and you only earn them if you:

1. Name the test. "By the ratio test..." or "By limit comparison with the series 1/n^2..."
2. Verify its conditions. Alternating series test: state that the terms are decreasing AND that their limit is 0. Integral test: state that f is positive, continuous, and decreasing. Comparison: state that the terms are positive.
3. Show the computation — the limit, the integral, whatever the test needs.
4. State the conclusion in words, tied back to the test you named.

Skipping step 2 is the classic BC error: a flawless ratio calculation with no named test and no conditions can score less than half. Treat every convergence question like a short proof, not a yes/no answer.

Worked example: the ratio test, step by step

Determine whether the sum from n = 1 to infinity of 3^n / n! converges.

• Step 1 — read the shape. There's a 3^n (something to the n-th power) and an n! (a factorial). Both scream ratio test.
• Step 2 — write the terms. a_n = 3^n / n! and a_(n+1) = 3^(n+1) / (n+1)!.
• Step 3 — set up the ratio. L = lim (n to infinity) of | a_(n+1) / a_n | = lim of [ 3^(n+1) / (n+1)! ] times [ n! / 3^n ].
• Step 4 — simplify. 3^(n+1) / 3^n = 3, and n! / (n+1)! = 1/(n+1). So the whole thing collapses to 3/(n+1).
• Step 5 — take the limit. lim (n to infinity) of 3/(n+1) = 0, so L = 0.
• Step 6 — apply the rule and conclude. Since L = 0 < 1, by the ratio test the series converges (in fact, absolutely).

Steps 1 and 6 are the ones examiners actually pay for. To check your own working on a nastier one, drop the series into the Math solver and read how it justifies each line, not just the final verdict.

A BC bonus: the alternating series error bound

If an alternating series passes the alternating series test, the error from stopping at the n-th partial sum is no bigger than the first term you left out, |a_(n+1)|. It's a recurring FRQ favourite ("how many terms guarantee an error under 0.01?") and basically free marks once you know it.

Test yourself

1. Does the sum from n = 1 to infinity of n / (2n + 1) converge or diverge?
2. Does the sum from n = 1 to infinity of 1 / n^(3/2) converge or diverge, and by which test?
3. Does the sum from n = 1 to infinity of (-1)^n / n converge? Absolutely or conditionally?

Answers:

1. Diverges by the nth-term test: lim n/(2n+1) = 1/2, which is not 0.
2. Converges by the p-series test, since p = 3/2 > 1.
3. Converges conditionally. It passes the alternating series test (the terms decrease to 0), but the absolute version, sum 1/n, is the harmonic series and diverges — so the convergence is conditional, not absolute.

Want a full set marked with the justification checked line by line? Generate practice in the Quiz maker, or paste your working into the Math solver for step-by-step feedback.

FAQ

Which convergence test should I try first?

Always the nth-term test — it's fast and can end the question on the spot. If the terms don't go to 0, the series diverges; if they do, it's inconclusive and you move to shape-matching.

When do I use the ratio test versus the comparison test?

Ratio test for factorials and n-th powers (2^n, n^n), because they cancel cleanly inside a ratio. Comparison — usually limit comparison — for rational or algebraic terms where you can spot a p-series with the same growth rate.

Is the root test on the AP Calculus BC exam?

No. It's a valid test and appears in many textbooks, but it isn't in the College Board course framework, so the ratio test covers everything you'll be asked. Spend your time on the eight tests that are actually on the exam.

Why did I lose marks when my answer was correct?

Almost always because you didn't name the test or verify its conditions. On FRQs the conclusion is one point and the rest are justification, so build the habit of writing "By the [test], since [conditions hold]..." every single time.

In short: Don't hunt for the right convergence test — read the shape of the terms, run the nth-term test first, and always name your test and check its conditions, because on BC that justification is where most of the marks quietly live.