Short on time? This page is the recap. For the partial-sums graph and full worked examples, use the full study guide. BC-only material. Prerequisites (limits at infinity 1.15, L’Hospital’s Rule 4.7, increasing/decreasing 5.3, partial sums 10.1, nth term test 10.3) are on the Calculus BC roadmap.
Recap
- An alternating series has signs +, −, +, − … (or −, +, − …): Σ (−1)ⁿ aₙ or Σ (−1)ⁿ⁺¹ aₙ with aₙ > 0.
- cos(nπ) = (−1)ⁿ, so it signals an alternating series. (−1)²ⁿ = 1 does not.
- Test: if aₙ₊₁ ≤ aₙ from some n on and lim aₙ = 0, the series converges.
- Why: each partial sum overshoots the last by less, so the zigzag of partial sums closes onto a single limit.
Key relationships
| Situation | What you can conclude | Test to name |
|---|---|---|
| aₙ decreasing (eventually) and aₙ → 0 | Converges | Alternating series test |
| lim aₙ ≠ 0 (or does not exist) | Diverges | nth term test |
| aₙ → 0 but not decreasing | No conclusion from this test | Try another method |
| Converges by this test | Σ aₙ may still diverge | Check separately (Topic 10.9) |
How to show aₙ decreases
| Method | Example |
|---|---|
| Compare denominators | 1/(2n + 9) gets smaller as the denominator grows |
| Derivative of f(x) with f(n) = aₙ | f′(x) < 0 for x ≥ N gives decreasing for n ≥ N |
| Ratio | aₙ₊₁/aₙ ≤ 1 |
Assumptions
- The test looks only at the positive size aₙ, never the signed term.
- Finitely many early terms do not matter, so “decreasing for n ≥ N” is enough. Say what N is.
Mistakes to avoid
- Checking only one condition. You need both, stated separately.
- “Diverges by the alternating series test.” Impossible: use the nth term test.
- Proving “decreasing” from a short list of terms. Give an inequality or a derivative.
- Missing the early rise. Terms such as √n/(n + 5) rise before they fall; state where the fall starts.
- Forgetting to conclude. End with “converges by the alternating series test”.
Quick self-check
- Does Σ (−1)ⁿ/n^(1/4) converge? (Yes: 1/n^(1/4) is positive, decreasing and tends to 0.)
- Does Σ (−1)ⁿ (n + 1)/n converge? (No: aₙ → 1, so it diverges by the nth term test. The alternating series test gives no conclusion.)
- Is Σ cos(nπ)/(n² + 1) alternating, and does it converge? (Yes and yes: cos(nπ) = (−1)ⁿ, and 1/(n² + 1) decreases to 0.)
Next: practice questions.