Top AP Calculus BC Series and Sequences Practice Problems

Introduction: Why Sequences and Series Matter in AP Calculus BC

If you’re preparing for AP Calculus BC, you already know that sequences and series make up a huge portion of the test. In fact, many students say that this unit feels like “a whole new class” compared to AP Calculus AB.

Why? Because the topics are abstract, require strong algebra skills, and demand careful attention to convergence tests, Taylor polynomials, and power series.

This guide will give you:

A breakdown of the most important series and sequences problems for AP Calculus BC.
Step-by-step practice problems with worked solutions.
Strategies to avoid the most common mistakes.
The best way to practice using RevisionDojo’s curated problem sets.

By the end, you’ll feel confident taking on even the toughest BC exam questions.

Key Topics You Must Know for Series and Sequences

Sequences

Definition of a sequence as a function from the natural numbers.
Limit of a sequence and convergence vs divergence.
Monotone and bounded sequences.

Series

Partial sums and convergence.
Geometric series and the formula S = a / (1 - r).
Harmonic series and why it diverges.
Alternating series and the Alternating Series Test.
Absolute vs conditional convergence.

Convergence Tests

Divergence Test (nth term test)
Ratio Test
Root Test
Integral Test
Comparison and Limit Comparison Tests

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.