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ENDURING UNDERSTANDING
LIM-7 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 LIM-7.A Determine whether a series converges or diverges. BC ONLY
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LIM-7.A.1 The nth partial sum is defined as the sum of the first n terms of a series. BC ONLY |
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. BC ONLY |
Blog Posts
Everyday series The most familiar series: Numbers
A Lesson on Sequences Suitable for Algebra 1 or as an introduction to series in calculus
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 LIM-7.A Determine whether a series converges or diverges. BC ONLY
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LIM-7.A.3 A geometric series is a series with a constant ratio between successive terms. BC ONLY |
LIM-7.A.4 If a is a real number and r is a real number such that |r| <1, then the geometric series |
Blog Posts
Geometric Series – Far Out A very interesting and instructive mistake
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10.3 The nth Term Test for Divergence BC ONLY
LEARNING OBJECTIVE LIM-7.A Determine whether a series converges or diverges. BC ONLY |
LIM-7.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.
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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10.4 The Integral Test for Convergence BC ONLY
LEARNING OBJECTIVE LIM-7.A Determine whether a series converges or diverges. BC ONLY |
LIM-7.A.6 The integral test is a method to determine whether a series converges or diverges. BC ONLY |
Blog Posts
Find post on Integral test in Good Questions?
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10.5 Harmonic Series and p-Series BC ONLY
LEARNING OBJECTIVE LIM-7.A Determine whether a series converges or diverges. |
LIM-7.A.7 In addition to geometric series, common series of numbers include the harmonic series, the alternating harmonic series, and p-series. BC ONLY |
Blog Posts
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10.6 Comparison Tests for Convergence BC ONLY
LEARNING OBJECTIVE LIM-7.A Determine whether a series converges or diverges. BC ONLY |
LIM-7.A.8 The comparison test is a method to determine whether a series converges or diverges. BC ONLY |
LIM-7.A.9 The limit comparison test is a method to determine whether a series converges or diverges. BC ONLY |
Blog Posts
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10.7 Alternating Series Test for Convergence BC ONLY
LEARNING OBJECTIVE LIM-7.A Determine whether a series converges or diverges. BC ONLY |
LIM-7.A.10 The alternating series test is a method to determine whether an alternating series converges. BC ONLY |
Blog Posts
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10.8 Ratio Test for Convergence BC ONLY
LEARNING OBJECTIVE LIM-7.A Determine whether a series converges or diverges. BC ONLY |
ESSENTIAL KNOWLEDGE LIM-7.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.
Blog Posts
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10.9 Determining Absolute or Conditional Convergence BC ONLY
LEARNING OBJECTIVE LIM-7.A Determine whether a series converges or diverges. BC ONLY |
LIM-7.A.12 A series may be absolutely convergent, conditionally convergent, or divergent. BC ONLY |
LIM-7.A.13 If a series converges absolutely, then it converges. BC ONLY | |
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. BC ONLY |
Blog Posts
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10.10 Alternating Series Error Bound BC ONLY
LEARNING OBJECTIVE LIM-7.B Approximate the sum of a series. BC ONLY |
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. BC ONLY |
Blog Posts
Error Bounds The alternating series error bound, and the Lagrange error bound
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ENDURING UNDERSTANDING
LIM-8 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 LIM-8.A Represent a function at a point as a Taylor polynomial. BC ONLY LIM-8.B Approximate function values using a Taylor polynomial. BC ONLY |
LIM-8.A.1 The coefficient of the nth degree term in a Taylor polynomial for a function f centered at x = a is |
LIM-8.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 | |
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. 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 LIM-8.C Determine the error bound associated with a Taylor polynomial approximation. BC ONLY |
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. BC ONLY |
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. BC ONLY |
Blog Posts
Then there is this – Existence theorems
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 LIM-8.D Determine the radius of convergence and interval of convergence for a power series. BC ONLY |
LIM-8.D.1 A power series is a series of the form |
LIM-8.D.2 If a power series converges, it either converges at a single point or has an interval of convergence. BC ONLY | |
LIM-8.D.3 The ratio test can be used to determine the radius of convergence of a power series. BC ONLY | |
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. BC ONLY | |
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. BC ONLY | |
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. 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 LIM-8.E Represent a function as a Taylor series or a Maclaurin series. BC ONLY LIM-8.F Interpret Taylor series and Maclaurin series. BC ONLY |
LIM-8.E.1 A Taylor polynomial for f (x) is a partial sum of the Taylor series for f(x). BC ONLY |
LIM-8.F.1 The Maclaurin series for |
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The Maclaurin series for sin(x), cos(x), and ex 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 LIM-8.G Represent a given function as a power series. BC ONLY |
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). BC ONLY |
Blog Posts
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REVIEW NOTES Type 10: Sequence and Series Questions (4-6-2018) A summary for reviewing sequences and series.
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