• 78 questions

AP Calculus Series Practice Test

Unofficial practice on infinite series for Unit 10 of AP Calculus BC: convergence tests, error bounds, Taylor and power series.

What does it mean for an infinite series to converge?

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Unofficial practice questions, not affiliated with the College Board. Bank checked October 13, 2026.

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Topics 10.1 to 10.3: Convergence, geometric series and the nth term test 14

  • For which values of the ratio r does the geometric series a + ar + ar² + … converge, if a is not 0? Wikipedia, "Geometric series"; course topic 10.2
  • If a series with terms aₙ converges, what must be true? Wikipedia, "Nth-term test"; course topic 10.3
  • The nth partial sum of a series is Sₙ = 2n/(n + 1). What is the sum of the series? Wikipedia, "Series (mathematics)"; course topic 10.1
  • The repeating decimal 0.777… equals 7/10 + 7/100 + 7/1000 + …. What fraction is it? Wikipedia, "Geometric series"; course topic 10.2
  • The series 1 − 1 + 1 − 1 + … has partial sums 1, 0, 1, 0, … What does that show? Wikipedia, "Grandi's series"; course topic 10.1
  • The terms of the series 1 + 1/√2 + 1/√3 + … approach 0. What does the nth term test conclude? Wikipedia, "Nth-term test"; course topic 10.3
  • What does it mean for an infinite series to converge? Wikipedia, "Series (mathematics)"; course topic 10.1
  • What does the nth term test show about the series whose terms are n/(2n + 1)? Wikipedia, "Nth-term test"; course topic 10.3
  • What is the sum of the geometric series 3 + 3/2 + 3/4 + 3/8 + …? Wikipedia, "Geometric series"; course topic 10.2
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Topics 10.1 to 10.3: Convergence, geometric series and the nth term test, continued

  • What is the sum of the series (1 − 1/2) + (1/2 − 1/3) + (1/3 − 1/4) + …? Wikipedia, "Telescoping series"; course topic 10.1
  • What is the sum of the series 2/3 + (2/3)² + (2/3)³ + …? Wikipedia, "Geometric series"; course topic 10.2
  • What is the sum of the series 4 − 2 + 1 − 1/2 + …? Wikipedia, "Geometric series"; course topic 10.2
  • Which geometric series diverges? Wikipedia, "Geometric series"; course topic 10.2
  • Which series does the nth term test show to diverge? Each sum starts at n = 1. Wikipedia, "Nth-term test"; course topic 10.3

Topics 10.4 to 10.6: Integral test, p-series and comparison tests 14

  • Does the series with terms 5/n^1.01, n ≥ 1, converge? Wikipedia, "Convergence tests"; course topic 10.5
  • For n ≥ 3, ln n > 1. What does that show about the series with terms (ln n)/n? Wikipedia, "Direct comparison test"; course topic 10.6
  • For which values of p does the p-series with terms 1/n^p converge? Wikipedia, "Convergence tests"; course topic 10.5
  • For x ≥ 2, an antiderivative of 1/(x ln x) is ln(ln x). What does the integral test conclude about the series with terms 1/(n ln n), n ≥ 2? Wikipedia, "Integral test for convergence"; course topic 10.4
  • The improper integral of 1/(x² + 1) from 1 to infinity equals π/4. What does the integral test conclude about the series with terms 1/(n² + 1), n ≥ 1? Wikipedia, "Integral test for convergence"; course topic 10.4
  • The integral test compares a series with terms f(n) to the improper integral of f. Under which conditions on f do calculus textbooks state the test? Wikipedia, "Integral test for convergence"; course topic 10.4
  • The limit comparison test is applied to aₙ = (2n + 1)/(n³ + 4) with bₙ = 1/n². What is the limit of aₙ/bₙ, and what follows? Wikipedia, "Limit comparison test"; course topic 10.6
  • Suppose 0 ≤ aₙ ≤ bₙ for all n, and the series of bₙ diverges. What can be concluded about the series of aₙ? Wikipedia, "Direct comparison test"; course topic 10.6
  • To test the series with terms n·e^(−n²), the integral test uses the integral of x·e^(−x²) from 1 to infinity. What is its value? Wikipedia, "Integral test for convergence"; course topic 10.4
  • To what value does the alternating harmonic series 1 − 1/2 + 1/3 − 1/4 + … converge? Wikipedia, "Harmonic series (mathematics)"; course topic 10.5
  • Which comparison proves that the series with terms 1/(n² + 5) converges? Wikipedia, "Direct comparison test"; course topic 10.6
  • Which comparison settles the series with terms 1/(3n − 2), n ≥ 1? Wikipedia, "Limit comparison test"; course topic 10.6
  • Which series converges? Each sum starts at n = 1. Wikipedia, "Convergence tests"; course topic 10.5
  • Which statement about the harmonic series 1 + 1/2 + 1/3 + 1/4 + … is true? Wikipedia, "Harmonic series (mathematics)"; course topic 10.5

Topics 10.7 to 10.9: Alternating series, ratio test and absolute convergence 14

  • The alternating series test applies to a series with terms (−1)ⁿbₙ, where bₙ > 0. When does it prove convergence? Wikipedia, "Alternating series test"; course topic 10.7
  • Does the series with terms (−1)ⁿ/√n, n ≥ 1, converge? Wikipedia, "Alternating series test"; course topic 10.7
  • Does the series with terms (−1)ⁿ·n/(n + 1) converge? Wikipedia, "Alternating series test"; course topic 10.7
  • How does the alternating harmonic series, with terms (−1)ⁿ⁺¹/n, converge, if at all? Wikipedia, "Conditional convergence"; course topic 10.9
  • How does the series with terms (−1)ⁿ/n² converge, if at all? Wikipedia, "Absolute convergence"; course topic 10.9
  • If the series of |aₙ| converges, what follows for the series of aₙ? Wikipedia, "Absolute convergence"; course topic 10.9
  • The ratio test is applied to the series with terms 3ⁿ/n!. What is the limit of |aₙ₊₁/aₙ|, and what follows? Wikipedia, "Ratio test"; course topic 10.8
  • The ratio test is applied to the series with terms n/2ⁿ. What is the limit of |aₙ₊₁/aₙ|, and what follows? Wikipedia, "Ratio test"; course topic 10.8
  • The series with terms cos(n)/n³ has both positive and negative terms in no regular pattern. Does it converge? Wikipedia, "Absolute convergence"; course topic 10.9
  • What does it mean for a series with terms aₙ to be conditionally convergent? Wikipedia, "Conditional convergence"; course topic 10.9
  • What does the ratio test give for the p-series with terms 1/n²? Wikipedia, "Ratio test"; course topic 10.8
  • What does the ratio test show about the series with terms 2ⁿ/n²? Wikipedia, "Ratio test"; course topic 10.8
  • What does the ratio test show about the series with terms n!/5ⁿ? Wikipedia, "Ratio test"; course topic 10.8
  • Which series meets both conditions of the alternating series test? Wikipedia, "Alternating series test"; course topic 10.7

Topics 10.10 and 10.12: Error bounds 8

  • A series passes the alternating series test, and a partial sum is used to estimate its sum. What does the alternating series error bound use as the bound? Wikipedia, "Alternating series"; course topic 10.10
  • A third-degree Maclaurin polynomial is used to estimate f(0.2). If the fourth derivative of f has absolute value at most 3 between 0 and 0.2, what is the Lagrange error bound? Wikipedia, "Taylor's theorem"; course topic 10.12
  • Every derivative of sin x has size at most 1. Which expression is the Lagrange error bound when the third-degree Maclaurin polynomial x − x³/6 estimates sin(0.1)? Wikipedia, "Taylor's theorem"; course topic 10.12
  • For the series 1 − 1/2! + 1/3! − 1/4! + …, what is the least number of terms for which the alternating series error bound guarantees an error below 0.001? Wikipedia, "Alternating series"; course topic 10.10
  • Let S be the sum of the alternating harmonic series and S₄ = 1 − 1/2 + 1/3 − 1/4. How do S₄ and S compare? Wikipedia, "Alternating series"; course topic 10.10
  • The Maclaurin series for cos x gives cos 1 = 1 − 1/2! + 1/4! − 1/6! + …, an alternating series with decreasing terms. If cos 1 is estimated by 1 − 1/2! + 1/4!, the error is at most what? Wikipedia, "Alternating series"; course topic 10.12
  • The sum of the series 1 − 1/4 + 1/9 − 1/16 + … is estimated by its first three terms. By the alternating series error bound, the error is at most what? Wikipedia, "Alternating series"; course topic 10.10
  • To bound the error of a third-degree Taylor polynomial at a point x, the Lagrange error bound needs the largest size of which derivative between the center and x? Wikipedia, "Taylor's theorem"; course topic 10.12

Topics 10.11 and 10.14: Taylor polynomials and Taylor series 13

  • A function has f(2) = 3, f′(2) = −1 and f″(2) = 4. What is its second-degree Taylor polynomial about x = 2? Wikipedia, "Taylor series"; course topic 10.11
  • The first-degree Taylor polynomial for √x about x = 4 is 2 + (x − 4)/4. What estimate does it give for √4.4? Wikipedia, "Taylor series"; course topic 10.11
  • For which x does the series 1 + x + x² + x³ + … equal 1/(1 − x)? Wikipedia, "Taylor series"; course topic 10.14
  • If the third derivative of f at 0 is 12, what is the coefficient of x³ in the Maclaurin polynomial for f? Wikipedia, "Taylor series"; course topic 10.11
  • The nth derivative of a function at 0 equals (n + 1)! for every n ≥ 0. What is its Maclaurin series? Wikipedia, "Taylor series"; course topic 10.14
  • The second-degree Maclaurin polynomial for cos x is 1 − x²/2. What estimate does it give for cos(0.2)? Wikipedia, "Taylor series"; course topic 10.11
  • The third-degree Taylor polynomial for f about x = 1 is 5 − 3(x − 1) + 2(x − 1)² + 4(x − 1)³. What is f″(1)? Wikipedia, "Taylor series"; course topic 10.11
  • What is the coefficient of x⁴ in the Maclaurin series for cos x? Wikipedia, "Taylor series"; course topic 10.14
  • What is the sum of the series 1 + 3 + 3²/2! + 3³/3! + …? Wikipedia, "Taylor series"; course topic 10.14
  • What is the third-degree Maclaurin polynomial for e^(2x)? Wikipedia, "Taylor series"; course topic 10.11
  • What is the third-degree Taylor polynomial for ln x about x = 1? Wikipedia, "Taylor series"; course topic 10.11
  • Which function has the Maclaurin series x − x³/3! + x⁵/5! − x⁷/7! + …? Wikipedia, "Taylor series"; course topic 10.14
  • Which is the Maclaurin series for e^x? Wikipedia, "Taylor series"; course topic 10.14

Topics 10.13 and 10.15: Power series and their intervals of convergence 15

  • A power series centered at x = 4 has radius of convergence 3. At which of these values of x must it converge? Wikipedia, "Power series"; course topic 10.13
  • A power series for f has radius of convergence 5. What is the radius of convergence of the series found by differentiating it term by term? College Board, AP Calculus AB and BC Course and Exam Description, essential knowledge LIM-8.D.6; course topic 10.13
  • For which x does the power series whose terms are n!·xⁿ converge? Wikipedia, "Radius of convergence"; course topic 10.13
  • For |x| < 1, which function does the series 1 + 2x + 3x² + 4x³ + … represent? Wikipedia, "Power series"; course topic 10.15
  • Integrating the series 1 − x² + x⁴ − x⁶ + … term by term from 0 to x gives the Maclaurin series of which function? Wikipedia, "Power series"; course topic 10.15
  • The series 1 + 2x + 4x² + 8x³ + … represents 1/(1 − 2x). For which x does it converge? Wikipedia, "Geometric series"; course topic 10.15
  • What are the first three nonzero terms of the Maclaurin series for e^(−x²)? Wikipedia, "Taylor series"; course topic 10.15
  • What are the first two nonzero terms of the Maclaurin series for sin(x²)? Wikipedia, "Taylor series"; course topic 10.15
  • What is the coefficient of x⁵ in the Maclaurin series for x²e^x? Wikipedia, "Taylor series"; course topic 10.15
  • What is the interval of convergence of the power series whose terms are (x + 2)ⁿ/(n·3ⁿ), n ≥ 1? Wikipedia, "Radius of convergence"; course topic 10.13
  • What is the interval of convergence of the power series whose terms are (x − 3)ⁿ/2ⁿ, n ≥ 0? Wikipedia, "Radius of convergence"; course topic 10.13
  • What is the interval of convergence of the power series x + x²/2 + x³/3 + x⁴/4 + …? Wikipedia, "Radius of convergence"; course topic 10.13
  • What is the radius of convergence of the power series whose terms are (x − 3)ⁿ/2ⁿ? Wikipedia, "Radius of convergence"; course topic 10.13
  • What is the radius of convergence of the power series whose terms are xⁿ/n!? Wikipedia, "Radius of convergence"; course topic 10.13
  • Which power series represents 1/(1 + x²) for |x| < 1? Wikipedia, "Geometric series"; course topic 10.15

How this list was compiled: Our own questions, written to Unit 10 (Infinite Sequences and Series; topics 10.1 to 10.15, all BC only) of the College Board's AP Calculus AB and BC Course and Exam Description, https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf, read 2026-10-13. Every answer is mathematics worked from what the question prints, and the computed ones were worked out a second time with computer algebra on 2026-10-13. The definition, test or theorem behind each answer was read on 2026-10-13 in the Wikipedia article or the essential knowledge statement named under the question. Licence: Own work.

How it works

This AP Calculus series practice test has 78 questions of our own on Unit 10 of AP Calculus BC, Infinite Sequences and Series. It is unofficial: this site is not affiliated with the College Board, the owner of AP.

Read the full guide

What Unit 10 covers

The College Board's course description, read October 13, 2026, gives Unit 10 fifteen topics and no weights among them, so the spread is ours. Six sets follow the topics in order: convergence, geometric series and the nth term test (14 questions); the integral test, p-series and comparison tests (14); alternating series, the ratio test and absolute convergence (14); error bounds (8); Taylor polynomials and series (13); power series (15).

Which tests the exam uses

The description names the tests the BC exam assesses: the nth term test for divergence and the integral, comparison, limit comparison, alternating series and ratio tests. Other methods are not assessed, so the root test does not appear here.

  • Try the nth term test first. If the terms do not approach 0, the series diverges and you are done.
  • After the ratio test gives a radius, test both endpoints separately. The ratio test says nothing there.
  • Know four series cold: 1/(1 − x), e^x, sin x and cos x. Most others come from them by substitution, differentiation or integration.

Questions and answers

What is Unit 10 in AP Calculus BC?

Infinite Sequences and Series, the last unit of the course. Its 15 topics run from the definition of convergence (10.1) to representing functions as power series (10.15).

How much of the AP Calculus BC exam is series?

From 15 to 20% of the multiple-choice section, by the College Board's course page. Only Unit 6, on integration, is weighted as heavily.

Are series on the AP Calculus AB exam?

No. Every topic of Unit 10 is marked BC only in the course description. The AB course ends with Unit 8, Applications of Integration.

Is this the AP Calculus BC Unit 10 progress check?

No. The College Board's Progress Check for Unit 10, about 45 multiple-choice and 3 free-response questions by the course description, is assigned by teachers in AP Classroom. These questions are our own.

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