# 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.

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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 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

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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.

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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

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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

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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

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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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 10.14 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

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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

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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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