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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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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 BC ONLY |
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10.3 The n^{th} 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_______________________________________________________________________________________________
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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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 |
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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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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 . 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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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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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 , where n is a non-negative integer, 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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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 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 |
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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._______________________________________________________________________________________________