Sequences and series is one of the most challenging — and most uniquely BC — portions of the AP Calculus BC exam. Unlike most calculus topics, which build directly on BC Calculus AB material, series introduces entirely new machinery: convergence tests, power series, and Taylor and Maclaurin series. Students who have not started reviewing series early often find themselves cramming the night before the exam, which is a frustrating and ineffective strategy for this topic.
This guide provides a focused first review of sequences and series for AP Calculus BC students, organized around the most commonly tested ideas, with practice problems and a two-week study plan.
ViewMath is not affiliated with or endorsed by the College Board or the AP program. Exam format details are based on publicly available AP Calculus BC information from the College Board. For official exam details, visit apcentral.collegeboard.org.
What the AP Calculus BC Exam Tests on Series
The AP Calculus BC exam (3 hours 15 minutes; 45 multiple-choice + 6 free-response questions) includes sequences and series as a BC-only topic. According to the College Board’s publicly available course framework, series topics appear across both the multiple-choice and free-response sections. The free-response section frequently includes at least one question that involves Taylor or Maclaurin series, and multiple-choice questions test convergence tests and interval of convergence.
Part 1: Sequences
A sequence is an ordered list of numbers, often defined by an explicit formula or a recursive rule.
- Convergence of a sequence: A sequence {aₙ} converges if limn→∞ aₙ = L for some finite L; otherwise it diverges.
- Squeeze Theorem for sequences: If bₙ ≤ aₙ ≤ cₙ and both {bₙ} and {cₙ} converge to L, then {aₙ} → L.
Practice: Does the sequence aₙ = (2n + 1)/(3n − 2) converge? Divide numerator and denominator by n: limn→∞ (2 + 1/n)/(3 − 2/n) = 2/3. (Answer: Yes, converges to 2/3)
Part 2: Convergence Tests for Series
A series is the sum of the terms of a sequence: Σ aₙ. The key question for any series is whether this sum converges to a finite value or diverges.
Divergence Test (Test for Divergence)
If limn→∞ aₙ ≠ 0 (or does not exist), then Σ aₙ diverges. Important: if limn→∞ aₙ = 0, the test is inconclusive — the series may still converge or diverge.
Practice: Does Σ n/(2n + 1) converge? limn→∞ n/(2n+1) = 1/2 ≠ 0. By the Divergence Test, the series diverges. (Answer: Diverges)
Integral Test
If f(x) is positive, continuous, and decreasing on [1, ∞), and aₙ = f(n), then Σ aₙ and ∫1∞ f(x) dx either both converge or both diverge.
p-Series Test
The series Σ 1/nᵖ converges if p > 1 and diverges if p ≤ 1.
Practice: Does Σ 1/n³/² converge? p = 3/2 > 1. (Answer: Converges)
Comparison Test
Compare an unfamiliar series to a known series. If 0 ≤ aₙ ≤ bₙ and Σ bₙ converges, then Σ aₙ converges. If 0 ≤ bₙ ≤ aₙ and Σ bₙ diverges, then Σ aₙ diverges.
Limit Comparison Test
If limn→∞ aₙ/bₙ = c where 0 < c < ∞, then Σ aₙ and Σ bₙ either both converge or both diverge.
Ratio Test
Compute L = limn→∞ |an+1/aₙ|.
- If L < 1: series converges absolutely.
- If L > 1 (or ∞): series diverges.
- If L = 1: test is inconclusive.
The Ratio Test is especially useful for series involving factorials or exponentials.
Practice: Test Σ n!/3ⁿ for convergence. L = limn→∞ [(n+1)!/3^(n+1)] / [n!/3ⁿ] = lim (n+1)/3 = ∞ > 1. (Answer: Diverges)
Alternating Series Test (Leibniz Test)
An alternating series Σ (−1)ⁿ bₙ converges if (1) bₙ > 0 for all n, (2) bₙ is decreasing, and (3) limn→∞ bₙ = 0.
Alternating series error bound: the error of a partial sum approximation is less than the absolute value of the first omitted term.
Part 3: Absolute and Conditional Convergence
- A series Σ aₙ converges absolutely if Σ |aₙ| converges.
- A series converges conditionally if Σ aₙ converges but Σ |aₙ| diverges.
- Absolute convergence implies convergence.
Part 4: Power Series and Interval of Convergence
A power series centered at x = a takes the form Σ cₙ(x − a)ⁿ. The key concepts are:
- Radius of convergence R: The series converges absolutely for |x − a| < R and diverges for |x − a| > R.
- Finding R using the Ratio Test: Compute limn→∞ |cn+1/cₙ| × |x − a|. Set this < 1 and solve for |x − a|.
- Interval of convergence: After finding R, check the endpoints x = a ± R separately (the Ratio Test is inconclusive at endpoints).
Part 5: Taylor and Maclaurin Series
A Taylor series for f(x) centered at x = a is:
f(x) = Σ f⁽ⁿ⁾(a)/n! × (x − a)ⁿ = f(a) + f′(a)(x−a) + f″(a)/2! × (x−a)² + …
A Maclaurin series is a Taylor series centered at a = 0.
Key Maclaurin series to memorize:
- eˣ = 1 + x + x²/2! + x³/3! + … (converges for all x)
- sin x = x − x³/3! + x⁵/5! − … (converges for all x)
- cos x = 1 − x²/2! + x⁴/4! − … (converges for all x)
- 1/(1 − x) = 1 + x + x² + x³ + … (converges for |x| < 1)
- ln(1 + x) = x − x²/2 + x³/3 − … (converges for −1 < x ≤ 1)
Practice: Write the first four terms of the Maclaurin series for eˣ². Substitute x² for x in the eˣ series: 1 + x² + x⁴/2! + x⁶/3! = 1 + x² + x⁴/2 + x⁶/6. (Answer: 1 + x² + x⁴/2 + x⁶/6 + …)
Practice: Find the 4th-degree Taylor polynomial for f(x) = sin x centered at x = 0. Using the Maclaurin series: P₄(x) = x − x³/6. (Answer: x − x³/6 — the x⁴ term has coefficient 0)
Two-Week AP Calculus BC Series Study Plan
Week 1: Convergence Tests
Days 1–2: Divergence Test, p-series, Integral Test. Practice identifying which test applies and applying it. Drill 8–10 problems per session.
Days 3–4: Comparison Test, Limit Comparison Test, and Ratio Test. Focus on series with factorials and n-th power terms.
Day 5: Alternating Series Test and absolute/conditional convergence. Mixed review of all 7 tests — 15 problems with a focus on choosing the right test.
Week 2: Power Series and Taylor/Maclaurin
Days 6–7: Radius and interval of convergence for power series. Practice 6–8 problems, including endpoint checks.
Days 8–9: Taylor and Maclaurin series. Memorize the key series above. Practice writing series from scratch using the derivative formula and by substitution into known series.
Day 10: Mixed free-response style problems. Work through 2–3 multi-part problems involving series error bounds, power series representations of functions, and interval of convergence.
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