Short on time? This page is the recap. For the rectangle diagram and worked examples, use the full study guide. BC-only material. Prerequisites (decreasing functions 5.3, substitution 6.9, parts 6.11, improper integrals 6.13, nth term test 10.3) are on the Calculus BC roadmap.
Recap
- Write aₙ = f(n). Draw width-1 rectangles of height aₙ.
- Rectangles below the curve: a₂ + … + aₙ ≤ ∫ from 1 to ∞ of f. A finite integral caps the partial sums, so the series converges.
- Rectangles above the curve: a₁ + … + aₙ ≥ ∫ from 1 to n + 1 of f. An infinite integral drags the partial sums up without bound, so the series diverges.
- Use the test after the nth term test has been inconclusive (aₙ → 0).
Key relationships
| Item | Statement | Note |
|---|---|---|
| Conditions | f positive, continuous, decreasing for x ≥ N | “Decreasing” usually from f′(x) < 0 |
| Conclusion | ∑ from N to ∞ of aₙ and ∫ from N to ∞ of f(x) dx both converge or both diverge | Never gives the sum |
| Improper integral | ∫ from N to ∞ of f = lim (b → ∞) of ∫ from N to b of f | Show the limit |
| Starting point | Finitely many terms do not affect convergence | Choose N where f starts decreasing |
| Background bound | ∫ from 1 to ∞ of f ≤ ∑ from 1 to ∞ of aₙ ≤ a₁ + ∫ from 1 to ∞ of f | Not required; a useful check |
Model justification: “f(x) = … is positive, continuous and decreasing for x ≥ N because …. ∫ from N to ∞ of f(x) dx = lim (b → ∞) of … = … (converges / diverges). So by the integral test, ∑ aₙ converges / diverges.”
Mistakes to avoid
- Claiming the sum equals the integral.
- Leaving out the conditions, or not showing why f is decreasing.
- Requiring “decreasing” from n = 1 when it only starts later; or applying the test to a function that is never decreasing on any [N, ∞), such as (2 + cos x)/x².
- Using the test on terms that change sign.
- Writing an antiderivative but no limit as b → ∞.
- Arguing “f(x) → 0, so the integral converges”.
Quick self-check
- ∑ 3/(n + 2)²? (∫ from 1 to ∞ of 3/(x + 2)² dx = 1, finite: converges, but the sum is not 1)
- ∑ 1/(2n + 5)? (∫ from 1 to b = ½ ln((2b + 5)/7) → ∞: diverges)
- Why can’t you use the integral test on ∑ (−1)ⁿ/n²? (The terms are not all positive)
Next: practice questions.