Top AP Calculus BC Series and Sequences Practice Problems | RevisionDojo

In AP Calculus BC, series and sequences make up a significant portion of the exam. You’ll need to identify convergence or divergence, work with Taylor and Maclaurin series, and apply limit theorems.

This RevisionDojo guide gives you:

• The most common problem types
• Fully worked AP-style examples
• Strategies to spot the right test quickly

📚 Key Topics to Know

Before diving into practice problems, make sure you’re confident in:

• Sequence limits and the definition of convergence
• Series convergence tests (nth-term, geometric, p-series, comparison, ratio, root, alternating)
• Taylor and Maclaurin series generation and use
• Error bounds for approximations

🔍 AP-Style Practice Problems

1. Limit of a Sequence

Problem: Determine lim⁡n→∞3n2+15n2−4\lim_{n \to \infty} \frac{3n^2 + 1}{5n^2 - 4}.

Solution:
Divide numerator and denominator by n2n^2:

3+1n25−4n2→35\frac{3 + \frac{1}{n^2}}{5 - \frac{4}{n^2}} \to \frac{3}{5}

2. Geometric Series

Problem: Does the series ∑n=0∞(23)n\sum_{n=0}^\infty \left( \frac{2}{3} \right)^n converge?

Solution:
It’s geometric with r=23r = \frac{2}{3}. Since ∣r∣<1|r| < 1, it converges to:

11−23=3\frac{1}{1 - \frac{2}{3}} = 3

3. p-Series Test

Problem: Determine if ∑n=1∞1n3/2\sum_{n=1}^\infty \frac{1}{n^{3/2}} converges.

Solution:
p-series with p=3/2>1p = 3/2 > 1, so it converges.

4. Ratio Test

Problem: Does ∑n=1∞n!3n\sum_{n=1}^\infty \frac{n!}{3^n} converge?

Solution:

lim⁡n→∞(n+1)!/3n+1n!/3n=lim⁡n→∞n+13=∞>1\lim_{n \to \infty} \frac{(n+1)! / 3^{n+1}}{n! / 3^n} = \lim_{n \to \infty} \frac{n+1}{3} = \infty > 1

Therefore, it diverges.

5. Alternating Series Test

Problem: Does ∑n=1∞(−1)n+1n\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} converge?

Solution:

• Terms decrease in absolute value: 1n\frac{1}{n} decreases
• lim⁡n→∞1n=0\lim_{n \to \infty} \frac{1}{n} = 0
  So it converges (conditionally).

6. Taylor Series Expansion

Problem: Find the Maclaurin series for exe^x.

Solution:

ex=∑n=0∞xnn!e^x = \sum_{n=0}^\infty \frac{x^n}{n!}

⚠️ Common Mistakes on AP Exam

• Forgetting that the nth-term test can only show divergence, not convergence
• Misidentifying rr in geometric series when there’s an alternating sign
• Applying ratio/root test incorrectly to p-series
• Forgetting the interval of convergence for power series

📊 Practice Strategy from RevisionDojo

• Drill 2–3 problems from each test weekly
• Use past FRQs for Taylor series — they often combine convergence with approximation
• Practice identifying the correct convergence test in under 15 seconds

🧭 Final Advice from RevisionDojo

Series and sequences can feel formula-heavy, but the AP exam rewards pattern recognition.
When you quickly match a problem type to the correct test, you’ll save time and earn high-value points.