AP® Calculus BC review sheet from Aim for Five (aimforfive.com/calc-bc/units/10/10-3)
Unit 10 · Topic 10.3
BC only10.3 The nth Term Test for Divergence
BC only. If the terms of a series don't approach 0, the series diverges. If they do approach 0, this test tells you nothing, and you need a different test.
Key terms
- nth term test
- limit of the terms
- divergence
- inconclusive
The test
This whole unit is BC only. nth term test for divergence: if lim (n → ∞) aₙ ≠ 0, or the limit doesn't exist, then Σ aₙ diverges.
Why: if a series converges, its partial sums settle down to some S. Then aₙ = Sₙ − Sₙ₋₁ → S − S = 0. So convergence forces the terms to go to 0. If the terms don't go to 0, the series can't converge.
It only ever proves divergence
The test runs in one direction only. If lim aₙ = 0, the test is inconclusive. The series might converge (like Σ 1/n²) or diverge (like Σ 1/n). You'd need another test to decide.
That's why it's called the nth term test for divergence. Never write “the terms go to 0, so the series converges by the nth term test.” That's one of the most common errors in this unit, and it loses points.
| lim (n → ∞) aₙ | Conclusion from this test |
|---|---|
| a nonzero number | diverges |
| infinite, or doesn't exist | diverges |
| 0 | no conclusion; try another test |
Limits you'll need
- Rational functions: compare the leading terms. n/(2n + 1) → 1/2. n²/(n³ + 1) → 0.
- (1 + 1/n)ⁿ → e, not 1.
- Oscillating terms: (−1)ⁿ has no limit, so Σ (−1)ⁿ diverges.
- Exponentials beat powers: n³/2ⁿ → 0, but 2ⁿ/n³ → ∞, so Σ 2ⁿ/n³ diverges.
- cos(1/n) → cos 0 = 1, so Σ cos(1/n) diverges.
- If needed, use L'Hospital's Rule (4.7) on the related function of x.
The logic behind “one direction only”
The true statement is: if Σ aₙ converges, then aₙ → 0. Its contrapositive is equally true: if aₙ does not approach 0, then Σ aₙ diverges. That's the nth term test. The converse, “if aₙ → 0, then Σ aₙ converges,” is false, and the harmonic series Σ 1/n is the standard counterexample.
Use it first
Check the limit of the terms before you try anything else. It takes seconds, and if the limit isn't 0, you're done. This also applies to power series: at an endpoint of an interval of convergence (10.13), the nth term test sometimes settles it immediately. If it is 0, you've lost nothing and can move on to the integral, comparison, alternating series or ratio test.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Terms approach a nonzero number
Does Σ (n = 1 to ∞) n/(2n + 1) converge or diverge?
Show the solutionHide the solution
- Step 1: lim (n → ∞) n/(2n + 1) = 1/2 (divide top and bottom by n).
- Step 2: The limit isn't 0, so the series diverges by the nth term test.
Answer: Diverges (nth term test: the terms approach 1/2, not 0).
- Example 2
A limit that looks like 1
Does Σ (n = 1 to ∞) (1 + 1/n)ⁿ converge or diverge?
Show the solutionHide the solution
- Step 1: lim (n → ∞) (1 + 1/n)ⁿ = e ≈ 2.718, a well-known limit.
- Step 2: e ≠ 0, so the series diverges by the nth term test.
Answer: Diverges (the terms approach e).
- Example 3
Trap: limit 0 means no conclusion
A student writes: “lim (n → ∞) 1/√n = 0, so Σ (n = 1 to ∞) 1/√n converges by the nth term test.” What's wrong, and what's the right conclusion?
Show the solutionHide the solution
- Step 1: The nth term test can only show divergence. A limit of 0 gives no information.
- Step 2: Use another test. Σ 1/√n = Σ 1/n^(1/2) is a p-series with p = 1/2 ≤ 1 (10.5).
- Step 3: So the series diverges.
Answer: The test was misused; the series actually diverges (p-series with p = 1/2).
Common mistakes
- Concluding convergence from lim aₙ = 0.
- Taking the limit of the partial sums instead of the terms.
- Thinking (1 + 1/n)ⁿ → 1.
- Not naming the test when you use it.
On the exam
- Multiple-choice questions often ask which statement must be true. “If Σ aₙ converges, then lim aₙ = 0” is true. “If lim aₙ = 0, then Σ aₙ converges” is false.
- In free response, write: “Since lim aₙ = 1/2 ≠ 0, the series diverges by the nth term test.”
Connected topics
Videos
Check yourself
4 questions on 10.3 The nth Term Test for Divergence. Pick an answer to see if you got it, and why.
For which of the following series does the nth term test show that the series diverges?
Let Σ (n = 1 to ∞) aₙ be a series. Which of the following statements must be true? I. If Σ aₙ converges, then lim (n→∞) aₙ = 0. II. If lim (n→∞) aₙ = 0, then Σ aₙ converges. III. If lim (n→∞) aₙ ≠ 0, then Σ aₙ diverges.
What does the nth term test show about the series Σ (n = 1 to ∞) (1 + 1/n)ⁿ?
Let aₙ = 2n/(n + 5) for n ≥ 1, and let Sₙ = a₁ + a₂ + ⋯ + aₙ. Which of the following statements are true? I. The sequence {aₙ} converges. II. The series Σ (n = 1 to ∞) aₙ converges. III. The sequence of partial sums {Sₙ} is unbounded.
0 of 4 answered