Short on time? This page is the recap. For the reasoning behind the test, the ratio graph and full worked examples, use the full study guide. BC-only material. Prerequisites (geometric series 10.2, nth term test 10.3, p-series 10.5, comparison tests 10.6) are on the Calculus BC roadmap.
Recap
- Compute L = lim (n → ∞) |aₙ₊₁/aₙ| (next term over current term).
- L < 1: converges (Σ |aₙ| converges too). L > 1 or L = ∞: diverges. L = 1: no conclusion.
- Idea: in the long run the series behaves like a geometric series with ratio L.
- Only the limit matters; early ratios can be anything.
- Why L > 1 means divergence: from some point on each term is bigger in size than the one before, so the terms cannot tend to 0 (nth term test).
- Why L < 1 means convergence: the tail is smaller than a convergent geometric series, so a comparison finishes the job.
Key relationships
| Algebra step | Result |
|---|---|
| (n + 1)!/n! | n + 1 |
| (2n + 2)!/(2n)! | (2n + 2)(2n + 1) |
| bⁿ⁺¹/bⁿ | b |
| (n + 1)ᵏ/nᵏ as n → ∞ | 1 |
| Polynomials of equal degree, ratio as n → ∞ | ratio of leading coefficients |
| Terms contain … | Use |
|---|---|
| n!, (2n)!, or powers like 3ⁿ | Ratio test |
| Only powers of n | p-series or comparison (ratio test gives L = 1) |
| Exactly a·rⁿ | Geometric series test (the ratio test also works) |
Assumptions
- Terms must be non-zero, so that the ratio exists.
- Keep the absolute value bars when terms can be negative, such as (−2)ⁿ.
Mistakes to avoid
- Upside-down ratio aₙ/aₙ₊₁, which gives 1/L.
- “L = 1 so it diverges.” No conclusion: Σ 1/n diverges, Σ 1/n² converges, and both have L = 1.
- Factorial slips: (n + 1)! = (n + 1) · n!, not n! + 1.
- Forgetting the exponential factor when cancelling, which turns L = 1/4 into L = 1.
- No conclusion sentence. Write: “L = … < 1, so the series converges by the ratio test.”
Quick self-check
- Σ 4ⁿ/n! (Ratio 4/(n + 1) → 0 < 1: converges.)
- Σ n³/2ⁿ (Ratio (n + 1)³/(2n³) → 1/2 < 1: converges.)
- Σ (n + 5)/n² (L = 1: no conclusion. Limit comparison with Σ 1/n shows it diverges.)
- Σ 2ⁿ/n² (Ratio 2n²/(n + 1)² → 2 > 1: diverges.)
Next: practice questions.