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ENDURING UNDERSTANDING
LIM7 Applying limits may allow us to determine the finite sum of infinitely many terms.
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Topic Name 
Essential Knowledge 
10.1 Defining Convergent and Divergent Infinite Series BC ONLY
LEARNING OBJECTIVE LIM7.A Determine whether a series converges or diverges. BC ONLY

LIM7.A.1 The nth partial sum is defined as the sum of the first n terms of a series. BC ONLY 
LIM7.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. BC ONLY 
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Everyday series The most familiar series: Numbers
Amortization An important use of a (finite) series – Find you mortgage payment without calculus.
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10.2 Working with Geometric Series BC ONLY
LEARNING OBJECTIVE LIM7.A Determine whether a series converges or diverges. BC ONLY

LIM7.A.3 A geometric series is a series with a constant ratio between successive terms. BC ONLY 
LIM7.A.4 If a is a real number and r is a real number such that r <1, then the geometric series BC ONLY 
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Geometric Series – Far Out A very interesting and instructive mistake
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10.3 The n^{th} Term Test for Divergence BC ONLY
LEARNING OBJECTIVE LIM7.A Determine whether a series converges or diverges. BC ONLY 
LIM7.A.5 The nth term test is a test for divergence of a series. BC ONLY 
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EXCLUSION STATEMENT 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.
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Convergence Test List A summary of the tests. Download and copy for your students (and yourself)
Which Convergence Test Should I Use? Part 1 You have a big choice
Which Convergence Test Should I Use? Part 2 Making the best choice
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10.4 The Integral Test for Convergence BC ONLY
LEARNING OBJECTIVE LIM7.A Determine whether a series converges or diverges. BC ONLY 
LIM7.A.6 The integral test is a method to determine whether a series converges or diverges. BC ONLY 
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Find post on Integral test in Good Questions?
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10.5 Harmonic Series and pSeries BC ONLY
LEARNING OBJECTIVE LIM7.A Determine whether a series converges or diverges. 
LIM7.A.7 In addition to geometric series, common series of numbers include the harmonic series, the alternating harmonic series, and pseries. BC ONLY 
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10.6 Comparison Tests for Convergence BC ONLY
LEARNING OBJECTIVE LIM7.A Determine whether a series converges or diverges. BC ONLY 
LIM7.A.8 The comparison test is a method to determine whether a series converges or diverges. BC ONLY 
LIM7.A.9 The limit comparison test is a method to determine whether a series converges or diverges. BC ONLY 
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10.7 Alternating Series Test for Convergence BC ONLY
LEARNING OBJECTIVE LIM7.A Determine whether a series converges or diverges. BC ONLY 
LIM7.A.10 The alternating series test is a method to determine whether an alternating series converges. BC ONLY 
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10.8 Ratio Test for Convergence BC ONLY
LEARNING OBJECTIVE LIM7.A Determine whether a series converges or diverges. BC ONLY 
ESSENTIAL KNOWLEDGE LIM7.A.11 The ratio test is a method to determine whether a series of numbers converges or diverges. BC ONLY 
EXCLUSION STATEMENT 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.
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10.9 Determining Absolute or Conditional Convergence BC ONLY
LEARNING OBJECTIVE LIM7.A Determine whether a series converges or diverges. BC ONLY 
LIM7.A.12 A series may be absolutely convergent, conditionally convergent, or divergent. BC ONLY 
LIM7.A.13 If a series converges absolutely, then it converges. BC ONLY  
LIM7.A.14 If a series converges absolutely, then any series obtained from it by regrouping or rearranging the terms has the same value. BC ONLY 
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10.10 Alternating Series Error Bound BC ONLY
LEARNING OBJECTIVE LIM7.B Approximate the sum of a series. BC ONLY 
LIM7.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. BC ONLY 
Blog Posts
Error Bounds The alternating series error bound, and the Lagrange error bound
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ENDURING UNDERSTANDING
LIM8 Power series allow us to represent associated functions on an appropriate interval.
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10.11 Finding Taylor Polynomial Approximations of Functions BC ONLY
LEARNING OBJECTIVES LIM8.A Represent a function at a point as a Taylor polynomial. BC ONLY LIM8.B Approximate function values using a Taylor polynomial. BC ONLY 
LIM8.A.1 The coefficient of the nth degree term in a Taylor polynomial for a function f centered at x = a is . BC ONLY 
LIM8.A.2 In many cases, as the degree of a Taylor polynomial increases, the nth degree polynomial will approach the original function over some interval. BC ONLY  
LIM8.B.1 Taylor polynomials for a function f centered at x = a can be used to approximate function values of f near x = a. BC ONLY 
Blog Posts
Introducing Power Series 1 Making better approximations
Introducing Power Series 2 Graphing and seeing the interval of convergence
Introducing Power Series 3 Questions pointing the way to power series
Graphing Taylor Polynomials Using a graphing calculator to graphs Taylor series
New Series from Old 1 Substituting
New Series from Old 2 Differentiating and Integrating
New Series from Old 3 Rational functions as geometric series
Good Question 16 What you get when you substitute.
Synthetic Summer Fun Finding the Taylor series coefficients without differentiating
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10.12 Lagrange Error Bound BC ONLY
LEARNING OBJECTIVE LIM8.C Determine the error bound associated with a Taylor polynomial approximation. BC ONLY 
LIM8.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. BC ONLY 
LIM8.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. BC ONLY 
Blog Posts
Error Bounds The alternating series error bound, and the Lagrange error bound
The Lagrange Highway a metaphor for the error bound
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10.13 Radius and Interval of Convergence of a Power Series BC ONLY
LEARNING OBJECTIVE LIM8.D Determine the radius of convergence and interval of convergence for a power series. BC ONLY 
LIM8.D.1 A power series is a series of the form , where n is a nonnegative integer, is a sequence of real numbers, and r is a real number. BC ONLY 
LIM8.D.2 If a power series converges, it either converges at a single point or has an interval of convergence. BC ONLY  
LIM8.D.3 The ratio test can be used to determine the radius of convergence of a power series. BC ONLY  
LIM8.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. BC ONLY  
LIM8.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. BC ONLY  
LIM8.D.6 The radius of convergence of a power series obtained by termbyterm differentiation or termbyterm integration is the same as the radius of convergence of the original power series. BC ONLY 
Blog Posts
Convergence Test List A summary of the tests. Download and copy for your students (and yourself)
Which Convergence Test Should I Use? Part 1 You have a big choice
Which Convergence Test Should I Use? Part 2 Making the best choice
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10.14 Finding Taylor or Maclaurin Series for a Function BC ONLY
LEARNING OBJECTIVES LIM8.E Represent a function as a Taylor series or a Maclaurin series. BC ONLY LIM8.F Interpret Taylor series and Maclaurin series. BC ONLY 
LIM8.E.1 A Taylor polynomial for f (x) is a partial sum of the Taylor series for f(x). BC ONLY 
LIM8.F.1 The Maclaurin series for is a geometric series. BC ONLY  
The Maclaurin series for sin(x), cos(x), and e^{x} provides the foundation for constructing the Maclaurin series for other functions. BC ONLY 
Blog Posts
Introducing Power Series 1 Making better approximations
Introducing Power Series 2 Graphing and seeing the interval of convergence
Introducing Power Series 3 Questions pointing the way to power series
Graphing Taylor Polynomials Using a graphing calculator to graphs Taylor series
New Series from Old 1 Substituting
New Series from Old 2 Differentiating and Integrating
New Series from Old 3 Rational functions as geometric series
Good Question 16 What you get when you substitute.
Synthetic Summer Fun Finding the Taylor series coefficients without differentiating
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10.15 Representing Functions as Power Series BC ONLY
LEARNING OBJECTIVE LIM8.G Represent a given function as a power series. BC ONLY 
LIM8.G.1 Using a known series, a power series for a given function can be derived using operations such as termbyterm differentiation or termbyterm integration, and by various methods (e.g., algebraic processes, substitutions, or using properties of geometric series). BC ONLY 
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REVIEW NOTES Type 10: Sequence and Series Questions (462018) A summary for reviewing sequences and series.
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