AP®︎ Calculus BC: Unit 10 Practice Test
Prepare for your quiz, test, or the AP exam with focused practice questions on Unit 10 of AP Calculus BC – Infinite Sequences and Series.
Questions List
Unit 10 (All Topics)
Question 1 Topic 10.1Easy
If for , what is ?
What You’re Being Tested On:
Explore the learning objectives taken directly from the College Board’s AP® Calculus BC Curriculum. Ensure you’re prepared for the exact topics covered on the AP® exam, in-class tests, and quizzes, and gain confidence in your mastery of the material.
Unit 10: Infinite Sequences and Series
Analyze infinite series, convergence tests, Taylor series, and error bounds.
Topic 10.1: Defining Convergent and Divergent Infinite Series
Learning Objective: LIM-7.A
Determine whether a series converges or diverges.
Essential Knowledge: LIM-7.A.1
The nth partial sum is defined as the sum of the first n terms of a series.
Essential Knowledge: LIM-7.A.2
An infinite series of numbers converges to a real number S (or has sum S), if and only if the limit of its sequence of partial sums exists and equals S.
Topic 10.2: Working with Geometric Series
Learning Objective: LIM-7.A
Determine whether a series converges or diverges.
Essential Knowledge: LIM-7.A.3
A geometric series is a series with a constant ratio between successive terms.
Essential Knowledge: LIM-7.A.4
If a is a real number and r is a real number such that |r| < 1, then the geometric series ∑_{n=0}^∞ ar^n = a/(1-r).
Topic 10.3: The nth Term Test for Divergence
Learning Objective: LIM-7.A
Determine whether a series converges or diverges.
Essential Knowledge: LIM-7.A.5
The nth term test is a test for divergence of a series.
Topic 10.4: Integral Test for Convergence
Learning Objective: LIM-7.A
Determine whether a series converges or diverges.
Essential Knowledge: LIM-7.A.6
The integral test is a method to determine whether a series converges or diverges.
Topic 10.5: Harmonic Series and p -Series
Learning Objective: LIM-7.A
Determine whether a series converges or diverges.
Essential Knowledge: LIM-7.A.7
In addition to geometric series, common series of numbers include the harmonic series, the alternating harmonic series, and p -series.
Topic 10.6: Comparison Tests for Convergence
Learning Objective: LIM-7.A
Determine whether a series converges or diverges.
Essential Knowledge: LIM-7.A.8
The comparison test is a method to determine whether a series converges or diverges.
Essential Knowledge: LIM-7.A.9
The limit comparison test is a method to determine whether a series converges or diverges.
Topic 10.7: Alternating Series Test for Convergence
Learning Objective: LIM-7.A
Determine whether a series converges or diverges.
Essential Knowledge: LIM-7.A.10
The alternating series test is a method to determine whether an alternating series converges.
Topic 10.8: Ratio Test for Convergence
Learning Objective: LIM-7.A
Determine whether a series converges or diverges.
Essential Knowledge: LIM-7.A.11
The ratio test is a method to determine whether a series of numbers converges or diverges. Exclusion: The nth term test for divergence, and the integral test, comparison test, limit comparison test, alternating series test, and ratio test for convergence are assessed on the AP Calculus BC Exam. Other methods are not assessed on the exam. However, teachers may include additional methods in the course, if time permits.
Topic 10.9: Determining Absolute or Conditional Convergence
Learning Objective: LIM-7.A
Determine whether a series converges or diverges.
Essential Knowledge: LIM-7.A.12
A series may be absolutely convergent, conditionally convergent, or divergent.
Essential Knowledge: LIM-7.A.13
If a series converges absolutely, then it converges.
Essential Knowledge: LIM-7.A.14
If a series converges absolutely, then any series obtained from it by regrouping or rearranging the terms has the same value.
Topic 10.10: Alternating Series Error Bound
Learning Objective: LIM-7.B
Approximate the sum of a series.
Essential Knowledge: LIM-7.B.1
If an alternating series converges by the alternating series test, then the alternating series error bound can be used to bound how far a partial sum is from the value of the infinite series.
Topic 10.11: Finding Taylor Polynomial Approximations of Functions
Learning Objective: LIM-8.A
Represent a function at a point as a Taylor polynomial.
Essential Knowledge: LIM-8.A.1
The coefficient of the nth degree term in a Taylor polynomial for a function f centered at x = a is f^(n)(a)/n!.
Essential Knowledge: LIM-8.A.2
In many cases, as the degree of a Taylor polynomial increases, the n-th degree polynomial will approach the original function over some interval.
Learning Objective: LIM-8.B
Approximate function values using a Taylor polynomial.
Essential Knowledge: LIM-8.B.1
Taylor polynomials for a function f centered at x = a can be used to approximate function values of f near x = a.
Topic 10.12: Lagrange Error Bound
Learning Objective: LIM-8.C
Determine the error bound associated with a Taylor polynomial approximation.
Essential Knowledge: LIM-8.C.1
The Lagrange error bound can be used to determine a maximum interval for the error of a Taylor polynomial approximation to a function.
Essential Knowledge: LIM-8.C.2
In some situations, the alternating series error bound can be used to bound the error of a Taylor polynomial approximation to the value of a function.
Topic 10.13: Radius and Interval of Convergence of Power Series
Learning Objective: LIM-8.D
Determine the radius of convergence and interval of convergence for a power series.
Essential Knowledge: LIM-8.D.1
A power series is a series of the form Σ_{n=0}^∞ a_n r^n, where n is a non-negative integer, {a_n} is a sequence of real numbers, and r is a real number.
Essential Knowledge: LIM-8.D.2
If a power series converges, it either converges at a single point or has an interval of convergence.
Essential Knowledge: LIM-8.D.3
The ratio test can be used to determine the radius of convergence of a power series.
Essential Knowledge: LIM-8.D.4
The radius of convergence of a power series can be used to identify an open interval on which the series converges, but it is necessary to test both endpoints of the interval to determine the interval of convergence.
Essential Knowledge: LIM-8.D.5
If a power series has a positive radius of convergence, then the power series is the Taylor series of the function to which it converges over the open interval.
Essential Knowledge: LIM-8.D.6
The radius of convergence of a power series obtained by term-by-term differentiation or term-by-term integration is the same as the radius of convergence of the original power series.
Topic 10.14: Finding Taylor or Maclaurin Series for a Function
Learning Objective: LIM-8.E
Represent a function as a Taylor series or a Maclaurin series.
Essential Knowledge: LIM-8.E.1
A Taylor polynomial for f(x) is a partial sum of the Taylor series for f(x).
Learning Objective: LIM-8.F
Interpret Taylor series and Maclaurin series.
Essential Knowledge: LIM-8.F.1
The Maclaurin series for 1/(1 - x) is a geometric series.
Essential Knowledge: LIM-8.F.2
The Maclaurin series for sin x, cos x, and e^x provides the foundation for constructing the Maclaurin series for other functions.
Topic 10.15: Representing Functions as Power Series
Learning Objective: LIM-8.G
Represent a given function as a power series.
Essential Knowledge: LIM-8.G.1
Using a known series, a power series for a given function can be derived using operations such as term-by-term differentiation or term-by-term integration, and by various methods (e.g., algebraic processes, substitutions, or using properties of geometric series).
