Short on time? This page is the recap. For the proof, the graph and worked examples, use the full study guide. BC-only material. Prerequisites (limits at infinity 1.15, L’Hospital’s Rule 4.7, partial sums 10.1, geometric series 10.2) are on the Calculus BC roadmap.
Recap
- A series has a sequence of terms aₙ and a sequence of partial sums Sₙ = a₁ + … + aₙ. It converges when Sₙ has a finite limit.
- Since aₙ = Sₙ − Sₙ₋₁, a convergent series must have aₙ → S − S = 0.
- So if aₙ does not approach 0, the series diverges. This is the nth term test.
- The test never proves convergence.
- Used forwards: if you know ∑ aₙ converges, you may use aₙ → 0 in a limit.
Key relationships
| Limit of the terms | Conclusion | Example |
|---|---|---|
| Non-zero number L | Diverges | ∑ 3n/(4n + 1): aₙ → 3/4 |
| ±∞ | Diverges | ∑ (1.02)ⁿ: aₙ → ∞ |
| Does not exist (oscillates) | Diverges | ∑ (−1)ⁿ n/(3n + 2): aₙ swings between about ±1/3 |
| 0 | No conclusion; use another method | 1 + ½ + ½ + ⅓ + ⅓ + ⅓ + …: terms → 0, but each block adds 1, so it diverges |
Model justification: “lim (n → ∞) of aₙ = … ≠ 0, so by the nth term test, ∑ aₙ diverges.”
Finding the limit
- Rational terms: compare the highest powers (equal powers → ratio of leading coefficients).
- 1^∞, ∞ · 0, 0/0 forms: rewrite with a continuous x and use logarithms or L’Hospital’s Rule. Example: (1 + 2/n)ⁿ → e².
- Terms with a factor (−1)ⁿ: check the size first. If the size approaches a non-zero number, the terms swing and the limit does not exist.
- The starting index and the first few terms never affect the limit.
Mistakes to avoid
- Writing “converges” because aₙ → 0.
- Saying the series “converges to L” because the terms approach L.
- Calling an oscillating limit “inconclusive”. If the limit does not exist, it is not 0: diverges.
- Treating (1 + k/n)ⁿ as → 1. It tends to eᵏ.
- Leaving out the limit value or the name of the test in a justification.
Quick self-check
- ∑ (4n + 1)/(9n − 2)? (aₙ → 4/9 ≠ 0, diverges)
- ∑ n²/(n³ + 1)? (aₙ → 0, nth term test inconclusive)
- ∑ cos(1/n)? (aₙ → cos 0 = 1 ≠ 0, diverges)
Next: practice questions.