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

Chapter 8-Infinite Series

Chapter 8-Infinite Series8.1 Series

The Definition of an Infinite Series

Convergence of Infinite Series

A Telescoping Series

The Harmonic Series

Basic Properties of Series

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?

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

The Divergence Test

EXAMPLE: What does the Divergence Test tell us about the geometric series What about the series ?

Series with Nonnegative Terms

EXAMPLE: Discuss convergence for the series

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.

The Integral Test

EXAMPLE: Show the following series converges and estimate its value

EXAMPLE: Show the following series converges and estimate its value .

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

Quick Quiz

Chapter 8-Infinite Series8.3 The Comparison Test

The Comparison Test for Convergence

EXAMPLE: For each of the following, determine whether the series converges or diverges.

The Limit Comparison Test

Quick Quiz

Chapter 8-Infinite Series8.4 Alternating Series

The Alternating Series Test

EXAMPLE: Analyze the series

EXAMPLE: Show that the following series converges

Absolute Convergence

THEOREM: If a series converges absolutely, then it converges.

EXAMPLE: Does the following series converge?

Conditional Convergence

Quick Quiz

Chapter 8-Infinite Series8.5 The Ratio and Root Tests

The Ratio Test

EXAMPLE: Apply the ratio test to the following

The Root Test

EXAMPLE: Apply the ratio test to the following

Quick Quiz

Chapter 8-Infinite Series8.6 Introduction to Power Series

EXAMPLE: Which of the following are power series in x?

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.

Power Series about an Arbitrary Base Point

EXAMPLE: Determine the interval of convergence for the following series.

Addition and Scalar Multiplication of Power Series

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).

Quick Quiz

Chapter 8-Infinite Series8.7 Representing Functions by Power Series

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).

The Relationship between the Coefficients and Derivatives

An Application to Differential Equations

EXAMPLE: Find a power series solution of the initial value problem dy/dx=y-x, y(0)=2.

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.

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.

Quick Quiz

Chapter 8-Infinite Series8.8 Taylor Series

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

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?

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

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.

Achieving a Desired Degree of Accuracy

EXAMPLE: Compute ln(1.2) to an accuracy of four decimals places.

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

Taylor Series Expansions of the Common Transcendental Functions

EXAMPLE: Show that

Using Taylor Series to Approximate

Quick Quiz

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

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