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Chapter 8-Infinite Series
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Chapter 8-Infinite Series8.1 Series
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
The Definition of an Infinite Series
Chapter 8-Infinite Series8.1 Series
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
The Definition of an Infinite Series
Chapter 8-Infinite Series8.1 Series
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Convergence of Infinite Series
Chapter 8-Infinite Series8.1 Series
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
A Telescoping Series
Chapter 8-Infinite Series8.1 Series
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
The Harmonic Series
Chapter 8-Infinite Series8.1 Series
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Basic Properties of Series
Chapter 8-Infinite Series8.1 Series
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Basic Properties of Series
Chapter 8-Infinite Series8.1 Series
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Series of Powers (Geometric Series)
Chapter 8-Infinite Series8.1 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 Series8.1 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 Series8.2 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 Series8.2 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 Series8.2 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,). Thenthe infinite series converges if and only if the improper integral converges.
Chapter 8-Infinite Series8.2 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 Series8.2 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 Series8.2 The Divergence Test and The Integral Test
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Quick Quiz
Chapter 8-Infinite Series8.3 The Comparison Test
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
The Comparison Test for Convergence
Chapter 8-Infinite Series8.3 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 Series8.3 The Comparison Test
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
The Limit Comparison Test
Chapter 8-Infinite Series8.3 The Comparison Test
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Quick Quiz
Chapter 8-Infinite Series8.4 Alternating Series
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
The Alternating Series Test
Chapter 8-Infinite Series8.4 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 Series8.4 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 Series8.4 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 Series8.4 Alternating Series
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Absolute Convergence
Chapter 8-Infinite Series8.4 Alternating Series
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Conditional Convergence
Chapter 8-Infinite Series8.4 Alternating Series
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Quick Quiz
Chapter 8-Infinite Series8.5 The Ratio and Root Tests
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
The Ratio Test
Chapter 8-Infinite Series8.5 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 Series8.5 The Ratio and Root Tests
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
The Root Test
Chapter 8-Infinite Series8.5 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 Series8.5 The Ratio and Root Tests
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Quick Quiz
Chapter 8-Infinite Series8.6 Introduction to Power Series
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Chapter 8-Infinite Series8.6 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 Series8.6 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 Series8.6 Introduction to Power Series
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Radius and Interval of Convergence
Chapter 8-Infinite Series8.6 Introduction to Power Series
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Radius and Interval of Convergence
Chapter 8-Infinite Series8.6 Introduction to Power Series
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Radius and Interval of Convergence
Chapter 8-Infinite Series8.6 Introduction to Power Series
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Radius and Interval of Convergence
Chapter 8-Infinite Series8.6 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
Chapter 8-Infinite Series8.6 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
Chapter 8-Infinite Series8.6 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 Series8.6 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 Series8.6 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 Series8.6 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 Series8.6 Introduction to Power Series
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Quick Quiz
Chapter 8-Infinite Series8.7 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 Series8.7 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 Series8.7 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 Series8.7 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 Series8.7 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 Series8.7 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 Series8.7 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
TN(c)=f(c), TN’(c)=f’(c), TN’’(c)=f’’(c), …, TN(N)(c)=f(N)(c)
EXAMPLE: Compute the Taylor polynomials of order one, two, and three for the function f(x)=e2x expanded with base point c=0.
Chapter 8-Infinite Series8.7 Representing Functions by Power Series
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Quick Quiz
Chapter 8-Infinite Series8.8 Taylor Series
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Taylor’s Theorem
f(x) = TN(x) + RN(x)THEOREM: For any natural number N, suppose that f is N + 1 times continuously differentiableon 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 Series8.8 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 T7 (x) with base point 0 of sin (x). If T7 (x) is used to approximate sin (x) for −1 ≤ x ≤ 1, what accuracy is guaranteed by Taylor’s Theorem?
Chapter 8-Infinite Series8.8 Taylor Series
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Estimating the Error TermTHEOREM: 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 Jx denote the closed interval with endpoints x and c. Thus, Jx = [c, x] if c ≤ x and Jx = [x, c] if x < c. Let
Then the error term RN(x) satisfies
Chapter 8-Infinite Series8.8 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 ex with base point 0 to approximate e-0.1. Estimate your accuracy.
Chapter 8-Infinite Series8.8 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 Series8.8 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 limN RN(x) exists, and
If and only if
Chapter 8-Infinite Series8.8 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
EXAMPLE: Show that
Chapter 8-Infinite Series8.8 Taylor Series
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Using Taylor Series to Approximate
Chapter 8-Infinite Series8.8 Taylor Series
Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved
Quick Quiz