Short on time? This page is the recap. For the flowchart, the proof and worked examples, use the full study guide. BC-only material. Prerequisites (Topics 10.3 to 10.8) are on the Calculus BC roadmap.
Recap
- Every series Σ aₙ has a partner Σ |aₙ| with all terms made positive.
- Absolutely convergent: Σ |aₙ| converges. Then Σ aₙ converges too.
- Conditionally convergent: Σ aₙ converges but Σ |aₙ| diverges. Convergence relies on cancellation.
- Divergent: Σ aₙ diverges (and so does Σ |aₙ|).
- If a series converges absolutely, regrouping or reordering its terms does not change its sum.
Order of testing
- Do the terms approach 0? If not, divergent (nth term test).
- Test Σ |aₙ| with a positive-term test: p-series, geometric, comparison, limit comparison, integral or ratio. Converges → absolutely convergent. Stop.
- If Σ |aₙ| diverges, test Σ aₙ itself, usually with the alternating series test. Converges → conditionally convergent; otherwise divergent.
Key relationships
| Situation | Conclusion | Why |
|---|---|---|
| Σ |aₙ| converges | Σ aₙ converges | 0 ≤ aₙ + |aₙ| ≤ 2|aₙ|, then comparison |
| Σ |aₙ| diverges | no conclusion yet | test Σ aₙ separately |
| Ratio test on |aₙ|: L < 1 | absolutely convergent | ratio test proves Σ |aₙ| converges |
| Ratio test on |aₙ|: L > 1 | divergent | |aₙ| grows, so aₙ does not → 0 |
| Σ (−1)ⁿ⁺¹/nᵖ, p > 1 | absolutely convergent | p-series converges |
| Σ (−1)ⁿ⁺¹/nᵖ, 0 < p ≤ 1 | conditionally convergent | p-series diverges; alternating series test |
| Σ (−1)ⁿ⁺¹/nᵖ, p ≤ 0 | divergent | terms do not → 0 |
Mistakes to avoid
- Reversing the theorem. “Σ aₙ converges, so Σ |aₙ| converges” is false.
- Stopping early. Σ |aₙ| diverging rules out absolute convergence only.
- Alternating series test on irregular signs (such as cos n or sin n in the numerator). Use absolute values and comparison instead.
- Missing conditions. For the alternating series test, show the terms decrease and approach 0.
- Missing absolute value bars in the ratio test.
- Reordering a conditionally convergent series. Its sum can change.
Quick self-check
- Classify Σ (−1)ⁿ/n³. (Absolutely convergent: p-series with p = 3 > 1.)
- Classify Σ (−1)ⁿ⁺¹/n^(1/3). (Conditionally convergent: p = 1/3 ≤ 1 so Σ |aₙ| diverges; the alternating series test gives convergence.)
- Classify Σ (−1)ⁿ 2ⁿ/n. (Divergent: the ratio of sizes approaches 2 > 1, so the terms do not approach 0.)
- Σ |aₙ| converges to 5. Can you reorder Σ aₙ without changing its sum? (Yes: it is absolutely convergent.)
Next: practice questions.