Unit 10 Infinite Sequences and Series   BC ONLY

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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

 

 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

 

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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

LIM-7.A Determine whether a series converges or diverges.  BC ONLY

 

 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 \sum\limits_{{n=0}}^{\infty }{{a{{r}^{n}}}}=\frac{a}{{1-r}}  BC ONLY

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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

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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

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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

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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

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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

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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 \frac{{{{f}^{{\left( n \right)}}}\left( a \right)}}{{n!}}. BC ONLY
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 \sum\limits_{{n=0}}^{\infty }{{{{a}_{n}}{{{\left( {x-r} \right)}}^{n}}}},  where n is a non-negative integer, \left\{ {{{a}_{n}}} \right\} is a sequence of real numbers, and r is a real number. BC ONLY
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 \frac{1}{{1-x}}  is a geometric series. BC ONLY
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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