Top AP Calculus BC Series and Sequences Practice Problems

Taylor and Maclaurin Series

• Taylor polynomial approximations.
• Maclaurin expansions for key functions: e^x, sin x, cos x, ln(1+x), (1+x)^n.
• Interval of convergence.

Practice Problem Set with Step-by-Step Solutions

Problem 1: Convergence of a Sequence

Determine whether the sequence aₙ = (3n + 1) / (2n + 5) converges.

Solution:
Take the limit as n → ∞. Divide numerator and denominator by n:
aₙ = (3 + 1/n) / (2 + 5/n) → 3/2.
Answer: The sequence converges to 3/2.

Problem 2: Geometric Series

Does the series Σ (1/3)^n from n = 0 to ∞ converge?

Solution:
This is a geometric series with a = 1 and r = 1/3.
Since |r| < 1, it converges to 1 / (1 - 1/3) = 3/2.
Answer: Converges to 3/2.

Problem 3: Alternating Series Test

Does the series Σ (-1)^n / n converge?

Solution:
Check conditions:

• Terms 1/n decrease.
• Limit of 1/n as n → ∞ = 0.
  So by the Alternating Series Test, it converges.
  Answer: Converges (conditionally, not absolutely).

Problem 4: Ratio Test

Does the series Σ (n! / 2^n) converge?

Solution:
Apply ratio test:
lim (aₙ₊₁ / aₙ) = ((n+1)! / 2^(n+1)) ÷ (n! / 2^n) = (n+1)/2.
As n → ∞, this → ∞, which is > 1.
Answer: Diverges.

Problem 5: Taylor Series Expansion

Find the Maclaurin series for e^x.

Solution:
Recall formula: e^x = Σ (x^n / n!) from n=0 to ∞.
Answer: 1 + x + x²/2! + x³/3! + … converges for all real x.

Problem 6: Interval of Convergence

Find the interval of convergence for the series Σ (x^n / n).

Solution:
Apply ratio test:
lim |(x^(n+1)/(n+1)) / (x^n/n)| = |x| * (n/(n+1)) = |x|.
So convergence for |x| < 1.
At x = 1, series = harmonic = diverges.
At x = -1, series = alternating harmonic = converges.
Answer: Interval is [-1, 1).

Common Mistakes Students Make

• Forgetting that the harmonic series diverges even though terms shrink.
• Misusing the ratio test when the limit equals 1 (it’s inconclusive).
• Forgetting to check endpoints when using ratio/root tests.
• Not knowing the basic Maclaurin series by heart (e^x, sin x, cos x, ln(1+x), geometric).
• Writing “converges” without specifying how or why (AP graders want justification).

How to Practice Sequences and Series Effectively

Memorization isn’t enough—you need to apply tests in exam-style problems.

This is why RevisionDojo is the best resource:

• Carefully designed AP-style sequences and series questions.
• Solutions that emphasize the reasoning the AP rubric expects.
• Guided practice on convergence tests.
• Timed drills to prepare you for the pressure of the exam.

RevisionDojo doesn’t just give you answers—it teaches you how to think like an AP grader.

Frequently Asked Questions

Q1: Do I need to memorize every convergence test?
Yes, but more importantly, you need to know when to apply each one.

Q2: What’s the difference between absolute and conditional convergence?

• Absolute: converges when all terms are positive.
• Conditional: converges only because of alternating signs.

Q3: How many Maclaurin series should I memorize?
At least: e^x, sin x, cos x, ln(1+x), (1+x)^n.

Q4: Do series questions appear in FRQs or just multiple choice?
Both. They often show up in FRQs as Taylor polynomials or convergence proofs.

Q5: Where can I get more practice problems?
The best option is RevisionDojo, which offers series/sequence sets crafted exactly like AP questions.

Conclusion: Conquer Sequences and Series for a 5

Sequences and series can feel intimidating, but with practice, they become some of the most predictable and rewarding points on the AP Calculus BC exam. By mastering convergence tests, Taylor expansions, and interval analysis, you’ll unlock one of the toughest sections of the test.

And with RevisionDojo’s practice problem sets, you can train with real AP-style questions until solving them feels natural. That’s the smartest path to a 5.