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Section 11.4 – Representing Functions with Power Series
4.Find the power series representation for centered about x = 0 and specify its radius of convergence. Infinite geometric with first term 1/3 and r = -2x/3 Converges when |r| < 1
8. Find the power series representation for centered about x = 0 and specify its radius of convergence. Radius of Convergence is 1
12. Use the power series representation of power series representation of the function to produce a
18.Find a power series representation of f(x) = ln x centered at x = 1. Specify the radius of convergence of the power series. Radius of convergence is 1
26.Use an appropriate identity to find the Maclaurin series for f(x) = sin x cos x
30. Given the function f defined by a.Find the first three nonzero terms in the Maclaurin series for the function f.
30. Given the function f defined by b. Find the first three terms in the Maclaurin series for the function g defined by
30. Given the function f defined by b. Find the first four terms in the Maclaurin series for the function h defined by
Section 11.4 – Representing Functions with Power Series 10.5.
Section 8.6/8.7: Taylor and Maclaurin Series Practice HW from Stewart Textbook (not to hand in) p. 604 # 3-15 odd, odd p. 615 # 5-25 odd, odd.
Power Series is an infinite polynomial in x Is a power series centered at x = 0. Is a power series centered at x = a. and.
Warm Up Determine the interval of convergence for the series:
S ECT. 9-5B M ORE T AYLOR S ERIES. Elementary Series.
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Taylor and Maclaurin Series Lesson Convergent Power Series Form Consider representing f(x) by a power series For all x in open interval I Containing.
Power Series Copyright © Cengage Learning. All rights reserved. 9.8.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 1.
In this section we develop general methods for finding power series representations. Suppose that f (x) is represented by a power series centered at.
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Section 9.2b. Do Now Find the fourth order Taylor polynomial that approximates near Before finding a bunch of derivatives, remember that we can use a.
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TAYLOR AND MACLAURIN how to represent certain types of functions as sums of power series You might wonder why we would ever want to express a known.
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Warm up Construct the Taylor polynomial of degree 5 about x = 0 for the function f(x)=e x. Graph f and your approximation function for a graphical.
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Sequences Definition - A function whose domain is the set of all positive integers. Finite Sequence - finite number of values or elements Infinite Sequence.
POWER SERIES Polynomial of degree n Example (Polynomial with infinit-degree) Power Series Example Polynomial with infinit-degree Power Series.
Ch 9.1 Power Series Calculus Graphical, Numerical, Algebraic by Finney, Demana, Waits, Kennedy.
Geometric Sequence – a sequence of terms in which a common ratio (r) between any two successive terms is the same. (aka: Geometric Progression) Section.
Find the local linear approximation of f(x) = e x at x = 0. Find the local quadratic approximation of f(x) = e x at x = 0.
What is the sum of the following infinite series 1+x+x 2 +x 3 +…x n … where 0
Representation of functions by Power Series We will use the familiar converging geometric series form to obtain the power series representations of some.
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Directions; Solve each integral below using one of the following methods: Change of Variables Geometric Formula Improper Integrals Parts Partial Fraction.
12. 1 A sequence is… (a) an ordered list of objects. (b) A function whose domain is a set of integers. Domain: 1, 2, 3, 4, …,n… Range a 1, a 2, a 3, a.
Do Now: Find both the local linear and the local quadratic approximations of f(x) = e x at x = 0 Aim: How do we make polynomial approximations for a given.
Taylor Series (4/7/06) We have seen that if f(x) is a function for which we can compute all of its derivatives (i.e., first derivative f '(x), second derivative.
Infinite Series 9 Copyright © Cengage Learning. All rights reserved.
Section 11.6 – Taylor’s Formula with Remainder. The Lagrange Remainder of a Taylor Polynomial where z is some number between x and c The Error of a Taylor.
The Ratio Test: Let Section 10.5 – The Ratio and Root Tests be a positive series and.
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Taylor Series (11/12/08) Given a nice smooth function f (x): What is the best constant function to approximate it near 0? Best linear function to approximate.
Sequences and Series It’s all in Section 9.4a!!!.
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MTH 253 Calculus (Other Topics) Chapter 11 – Infinite Sequences and Series Section 11.8 –Taylor and Maclaurin Series Copyright © 2009 by Ron Wallace, all.
Taylor and MacLaurin Series Lesson 8.8. Taylor & Maclaurin Polynomials Consider a function f(x) that can be differentiated n times on some interval I.
Final Review – Exam 4. Radius and Interval of Convergence (11.1 & 11.2) Given the power series, Refer to lecture notes or textbook of section 11.1 and.
Arithmetic and Geometric Means OBJ: Find arithmetic and geometric means.
Def: The power series centered at x = a: 1 x is the variable and the c’s are constants (coefficients) Lecture 29 – Power Series.
Taylor’s Theorem Section 9.3a. While it is beautiful that certain functions can be represented exactly by infinite Taylor series, it is the inexact Taylor.
Section 8.5: Power Series Practice HW from Stewart Textbook (not to hand in) p. 598 # 3-17 odd.
1 February 5 Complex numbers 2.1 Introduction 2.2 Real and imaginary parts of a complex number 2.3 The complex plane 2.4 Terminology and notation Solution.
OBJECTIVES: Determine if a Sequence is Geometric Find a Formula for a Geometric Sequence Find the Sum of a Geometric Sequence Find the Sum of a Geometric.
Resultant of two forces Resultant of parallel forces Resultant of perpendicular forces Resultant two forces at any angle Learning objectives.
1 Chapter 9. 2 Does converge or diverge and why?
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