Short on time? This page is the recap. For the four-case diagram, the reasons the tests work and worked examples, use the full study guide. BC-only material. Prerequisites (limits at infinity 1.15, geometric series 10.2, nth term test 10.3, p-series 10.5) are on the Calculus BC roadmap.
Recap
- Compare the series you are testing (aₙ) with a benchmark you know (bₙ): usually a p-series or a geometric series.
- Both tests need positive terms, at least from some n onwards.
- Direct comparison links the series with an inequality. Limit comparison links them with the limit of aₙ/bₙ.
- Make a guess first (converge or diverge?) from the dominant terms, then pick the comparison that can prove it.
Key relationships
| Test | Condition | Conclusion |
|---|---|---|
| Direct, convergence | 0 ≤ aₙ ≤ bₙ eventually, ∑ bₙ converges | ∑ aₙ converges |
| Direct, divergence | aₙ ≥ bₙ ≥ 0 eventually, ∑ bₙ diverges | ∑ aₙ diverges |
| Direct, other two cases | smaller than divergent, or bigger than convergent | No conclusion |
| Limit comparison | aₙ, bₙ > 0, aₙ/bₙ → L with 0 < L < ∞ | Both converge or both diverge |
Choosing bₙ
- Keep the fastest-growing term on top and on the bottom: (3n + 1)/(n³ − 2) → bₙ = 1/n².
- Roots count as powers: √n/(n² + 4) → bₙ = 1/n^(3/2).
- Exponentials beat powers: 2ⁿ/(5ⁿ + n²) → bₙ = (2/5)ⁿ.
Assumptions
- Inequalities only need to hold for n ≥ some N; say which N.
- Bounded pieces: |sin n| ≤ 1, 0 ≤ cos² n ≤ 1, 0 < arctan n < π/2; ln n > 1 for n ≥ 3.
Mistakes to avoid
- Useless direction, such as “smaller than ∑ 1/n, so it diverges”.
- No positivity check.
- lim aₙ instead of lim aₙ/bₙ.
- Accepting L = 0 or ∞ in the basic limit comparison test.
- Not saying why the benchmark converges or diverges (state p or the ratio).
Quick self-check
- Does ∑ 1/(n³ + n) converge? (Yes: 0 < aₙ ≤ 1/n³, and ∑ 1/n³ converges, p = 3.)
- Does ∑ (n + 1)/(n² + 1) converge? (No: limit comparison with 1/n gives L = 1, and ∑ 1/n diverges.)
- Does ∑ 1/(5ⁿ − 1) converge? (Yes: aₙ > (1/5)ⁿ so direct comparison fails, but limit comparison with (1/5)ⁿ gives L = 1, and the geometric series converges.)
Next: practice questions.