CMS Curriculum Guides 2011-2012
AP Calculus
Unit Title: Polynomial Approximations and Series
Suggested Time: 3 Weeks
Enduring understanding (Big Idea): Geometric series, convergence of series, tests for convergence of series, endpoint convergence,
representing functions by series, identifying a series, constructing a series, series for sin x and cos x, Maclaurin and Taylor series,
combining Taylor series, Taylor polynomials, remainder estimation theorem.
Essential Questions: what is a sequence? What is an infinite series? Properties of geometric series and finding the sum to infinity.
How to test the convergence of series. What is a power series? How can we represent functions in power series? What is Taylor
series? What is Taylor polynomial? How do we determine the convergence of Taylor series?
AP Topic
Textbook Correlation
Related AP
Questions
Resources
Concept of series.
 A series is defined as a sequence of
partial sums, and convergence is defined
in terms of the limit of the sequence of
partial sums.
 Technology can be used to explore
convergence and divergence.
Series of constants
 Motivating examples, including decimal
expansion.
 Geometric series with applications.
 The harmonic series.
 Alternating series with error bound.
 Terms of series as areas of rectangles and
their relationship to improper integrals,
including the integral test and its use in
testing the convergence of p-series.
8.1 Sequences
1971 BC 4, 1972 BC
4
Exercises from sections
8.1
9.4 Radius of convergence
1994 BC 5, 1992 BC
6, 1991 BC 5, 2002
BC 6, 2006 BC 6
Video lessons
9.5 Testing convergence at end
points
Worksheet on
convergence of series
CMS Curriculum Guides 2011-2012
AP Calculus

The ratio test for convergence and
divergence.
 Comparing series to test for convergence
or divergence.
Taylor series
 Taylor polynomial approximation with
graphical demonstration of convergence
 (for example, viewing graphs of various
Taylor polynomials of the sine function
 Approximating the sine curve).
 Maclaurin series and the general Taylor
series centered at x = a.
 Maclaurin series for the functions ex , sin
x, cos x, and
1
 1 x .
 Formal manipulation of Taylor series and
shortcuts to computing Taylor series,
including substitution, differentiation,
antidifferentiation and the formation of
new series from known series.
 Functions defined by power series.
 Radius and interval of convergence of
power series.
 Lagrange error bound for Taylor
polynomials.
9.1 Power series
9.2 Taylor series
9.3 Taylor’s theorem
9.5 Testing convergence at end
points
1997 BC 2, 1996 BC
2, 1995 BC 4, 1993
BC 5, 1990 BC 5,
2002 BC 6, 200 BC
3, 1999 BC 4, 1998
BC 3, 2008 BC 3,
2007 BC 6, 2005 BC
6, 2004 BC 6
Video lessons
Taylor Maclaurin series
millionaire
Review sheet on series
Exercises from
textbook sections 9.1,
9.2, 9.3 and 9.5
Prior Knowledge: Working with factorial notation, finding the nth term of sequences and series, working with sigma notation,
integrals and derivatives.
CMS Curriculum Guides 2011-2012
AP Calculus
Essential Terms Developed in This Unit
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Useful Terms Referenced in This Unit
Convergence of series
Absolute convergence
Harmonic series
Telescoping series
Taylor polynomial
Term by term integration
Interval of convergence
Resources
Additional Practice/Skills Worksheets
CMP2 Website –online & technology resources
Formal Assessment
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Check-Ups
Partner Quiz
Unit Test
Assessment Options
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Multiple-Choice
Question Bank
ExamView CD-ROM
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Geometric series
Terms Developed in Previous Unit
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Integration
Derivative