Chapter 8-Infinite Series Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved.

Chapter 8-Infinite Series 8.1 Series Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Series of Powers (Geometric Series) EXAMPLE: At a certain aluminum recycling plant, the recycling process turns n pounds of used aluminum into 9n/10 pounds of new aluminum. Including the initial quantity, how much usable aluminum will 100 pounds of virgin aluminum ultimately yield, if we assume that it is continually returned to the same recycling plant?

Chapter 8-Infinite Series 8.1 Series Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. Which of the numbers 9/8, 10/8, 11/8, 12/8, is a partial sum of 2. True or false: The sum of two convergent series is also convergent. 3. What is the value of 4. Does converge or diverge?

Chapter 8-Infinite Series 8.2 The Divergence Test and The Integral Test Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Divergence Test EXAMPLE: What does the Divergence Test tell us about the geometric series What about the series ?

Chapter 8-Infinite Series 8.2 The Divergence Test and The Integral Test Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Series with Nonnegative Terms EXAMPLE: Discuss convergence for the series

Chapter 8-Infinite Series 8.2 The Divergence Test and The Integral Test Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Integral Test THEOREM: Let f be a positive, continuous, decreasing function on the interval [1,  ). Then the infinite series converges if and only if the improper integral converges.

Chapter 8-Infinite Series 8.2 The Divergence Test and The Integral Test Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Integral Test EXAMPLE: Show the following series converges and estimate its value EXAMPLE: Show the following series converges and estimate its value.

Chapter 8-Infinite Series 8.2 The Divergence Test and The Integral Test Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved p-series THEOREM: Fix a real number p. The series converges if p>1 and diverges if p≤1. EXAMPLE: Determine whether the following series is convergent

Chapter 8-Infinite Series 8.3 The Comparison Test Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Comparison Test for Convergence EXAMPLE: For each of the following, determine whether the series converges or diverges.

Chapter 8-Infinite Series 8.4 Alternating Series Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Alternating Series Test EXAMPLE: Analyze the series EXAMPLE: Show that the following series converges

Chapter 8-Infinite Series 8.4 Alternating Series Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Absolute Convergence THEOREM: If a series converges absolutely, then it converges.

Chapter 8-Infinite Series 8.4 Alternating Series Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Absolute Convergence EXAMPLE: Does the following series converge?

Chapter 8-Infinite Series 8.5 The Ratio and Root Tests Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Ratio Test EXAMPLE: Apply the ratio test to the following

Chapter 8-Infinite Series 8.5 The Ratio and Root Tests Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Root Test EXAMPLE: Apply the ratio test to the following

Chapter 8-Infinite Series 8.6 Introduction to Power Series Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Which of the following are power series in x?

Chapter 8-Infinite Series 8.6 Introduction to Power Series Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Radius and Interval of Convergence THEOREM: Let be a power series. Then precisely one of the following statements holds: a) The series converges absolutely for every real x; b) There is a positive number R such that the series converges absolutely for |x| < R and diverges for |x| > R; c) The series converges only at x = 0.

Chapter 8-Infinite Series 8.6 Introduction to Power Series Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Power Series about an Arbitrary Base Point EXAMPLE: Determine the interval of convergence for the following series.

Chapter 8-Infinite Series 8.6 Introduction to Power Series Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Addition and Scalar Multiplication of Power Series

Chapter 8-Infinite Series 8.6 Introduction to Power Series Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Differentiation and Antidifferentiation of Power Series

Chapter 8-Infinite Series 8.6 Introduction to Power Series Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Differentiation and Antidifferentiation of Power Series EXAMPLE: Calculate the derivative and the indefinite integral of the power series below for x in the interval of convergence I=(-1,1).

Chapter 8-Infinite Series 8.7 Representing Functions by Power Series Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Power Series Expansions of Some Standard Functions

Chapter 8-Infinite Series 8.7 Representing Functions by Power Series Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Power Series Expansions of Some Standard Functions EXAMPLE: Express the following as a power series with base point 0. EXAMPLE: Find a power series representation for the function F(x)=ln(1+x).

Chapter 8-Infinite Series 8.7 Representing Functions by Power Series Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Relationship between the Coefficients and Derivatives

Chapter 8-Infinite Series 8.7 Representing Functions by Power Series Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved An Application to Differential Equations EXAMPLE: Find a power series solution of the initial value problem dy/dx=y-x, y(0)=2.

Chapter 8-Infinite Series 8.7 Representing Functions by Power Series Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Taylor Series and Polynomials DEFINITION: If a function f is N times continuously differentiable on a interval containing c, then is called the Taylor polynomial of order N and base point c for the function f. If f is infinitely differentiable, then we have a Taylor series.

Chapter 8-Infinite Series 8.7 Representing Functions by Power Series Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Taylor Series and Polynomials THEOREM: Suppose that f is N times continuously differentiable. Then T N (c)=f(c), T N ’(c)=f’(c), T N ’’(c)=f’’(c), …, T N (N) (c)=f (N) (c) EXAMPLE: Compute the Taylor polynomials of order one, two, and three for the function f(x)=e 2x expanded with base point c=0.

Chapter 8-Infinite Series 8.8 Taylor Series Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Taylor’s Theorem f(x) = T N (x) + R N (x) THEOREM: For any natural number N, suppose that f is N + 1 times continuously differentiable on an open interval I centered at c. If x is a point in I, then there is a number s between c and x such that

Chapter 8-Infinite Series 8.8 Taylor Series Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Taylor’s Theorem EXAMPLE: Calculate the order 7 Taylor polynomial T 7 (x) with base point 0 of sin (x). If T 7 (x) is used to approximate sin (x) for −1 ≤ x ≤ 1, what accuracy is guaranteed by Taylor’s Theorem?

Chapter 8-Infinite Series 8.8 Taylor Series Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Estimating the Error Term THEOREM: Let f be a function that is N + 1 times continuously differentiable on an open interval I centered at c. For each x in I let J x denote the closed interval with endpoints x and c. Thus, J x = [c, x] if c ≤ x and J x = [x, c] if x < c. Let Then the error term R N (x) satisfies

Chapter 8-Infinite Series 8.8 Taylor Series Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Estimating the Error Term EXAMPLE: Use the third order Taylor polynomial of e x with base point 0 to approximate e -0.1. Estimate your accuracy.

Chapter 8-Infinite Series 8.8 Taylor Series Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Achieving a Desired Degree of Accuracy EXAMPLE: Compute ln(1.2) to an accuracy of four decimals places.

Chapter 8-Infinite Series 8.8 Taylor Series Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Taylor Series Expansions of the Common Transcendental Functions THEOREM: Suppose that f is infinitely differentiable on an interval containing points c and x. Then exists if and only lim N  R N (x) exists, and If and only if

Chapter 8-Infinite Series 8.8 Taylor Series Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Taylor Series Expansions of the Common Transcendental Functions EXAMPLE: Show that

Chapter 8-Infinite Series Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved.

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Chapter 8-Infinite Series Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved.

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Presentation on theme: "Chapter 8-Infinite Series Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved."— Presentation transcript:

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