Chapter 9 Infinite Series.

Slides:

Advertisements

Similar presentations

Chapter 10 Infinite Series by: Anna Levina edited: Rhett Chien.

Advertisements

Assignment Answers: Find the partial sum of the following: 1. = 250/2 ( ) = 218, = 101/2 (1/2 – 73/4) = Find the indicated n th.

© 2010 Pearson Education, Inc. All rights reserved.

Copyright©2000 by Houghton Mifflin Company. All rights reserved. 1 QUESTION.

Infinite Series: “The Comparison, Ratio, and Root Tests”

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 9- 1.

Copyright © 2005 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

Calculus, 8/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Definition (p. 626)

Geometric Sequences and Series

AP Calculus Objective: Understand definition of convergent infinite series and use properties of infinite geometric series.

Limits and Their Properties

Chapter 1 Infinite Series. Definition of the Limit of a Sequence.

Chapter 1 Infinite Series, Power Series

1.5 Infinite Limits. Copyright © Houghton Mifflin Company. All rights reserved Figure 1.25.

Chapter Two Limits and Their Properties. Copyright © Houghton Mifflin Company. All rights reserved. 2 | 2 The Tangent Line Problem.

Chapter 1 Limits and Their Properties. Copyright © Houghton Mifflin Company. All rights reserved.1 | 2 Figure 1.1: Definition of the Slope of a Line.

9.1 Sequences. A sequence is a list of numbers written in an explicit order. n th term Any real-valued function with domain a subset of the positive integers.

Copyright © 2011 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 8 Sequences and Infinite Series.

Infinite Series Copyright © Cengage Learning. All rights reserved.

Chapter 15 Vector Analysis. Copyright © Houghton Mifflin Company. All rights reserved.15-2 Definition of Vector Field.

SERIES AND CONVERGENCE

Calculus, 9/E by Howard Anton, Irl Bivens, and Stephen Davis Copyright © 2009 by John Wiley & Sons, Inc. All rights reserved. Major theorems, figures,

Goal: Does a series converge or diverge? Lecture 24 – Divergence Test 1 Divergence Test (If a series converges, then sequence converges to 0.)

Infinite Sequences and Series

Infinite Series Copyright © Cengage Learning. All rights reserved.

Chapter 9.6 THE RATIO AND ROOT TESTS. After you finish your HOMEWORK you will be able to… Use the Ratio Test to determine whether a series converges or.

Copyright © Cengage Learning. All rights reserved.

One important application of infinite sequences is in representing “infinite summations.” Informally, if {a n } is an infinite sequence, then is an infinite.

Chapter 9 AP Calculus BC. 9.1 Power Series Infinite Series: Partial Sums: If the sequence of partial sums has a limit S, as n  infinity, then we say.

Advance Calculus Diyako Ghaderyan 1 Contents:  Applications of Definite Integrals  Transcendental Functions  Techniques of Integration.

MTH 253 Calculus (Other Topics)

Copyright © Cengage Learning. All rights reserved. 11 Infinite Sequences and Series.

Chapter 1 Limits and Their Properties. Copyright © Houghton Mifflin Company. All rights reserved.21-2 Figure 1.1.

MTH 253 Calculus (Other Topics) Chapter 11 – Infinite Sequences and Series Section 11.5 – The Ratio and Root Tests Copyright © 2009 by Ron Wallace, all.

Chapter 9 Infinite Series. 9.1 Sequences Warm Up: Find the next 3 terms… 1. 2, 6, 10, 14, … Common Diff: 4 18, 22, , 6, 12, 24, … Doubled Sequence.

A LESSON BY U S PRAJAPATI, PGT MATH, KV KHAGAUL GEOMETRIC SEQUENCES AND SERIES.

Logarithmic, Exponential, and Other Transcendental Functions

Final Review – Exam 3 Sequences & Series Improper Integrals.

Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions.

Lecture 6: Convergence Tests and Taylor Series. Part I: Convergence Tests.

Advance Calculus Diyako Ghaderyan 1 Contents:  Applications of Definite Integrals  Transcendental Functions  Techniques of Integration.

1 Chapter 9. 2 Does converge or diverge and why?

OBJECTIVE TSW (1) list the terms of a sequence; (2) determine whether a sequence converges or diverges; (3) write a formula for the nth term of a sequence;

Does the Series Converge?

9-6 The Ratio Test Rizzi – Calc BC. Objectives  Use the Ratio Test to determine whether a series converges or diverges.  Review the tests for convergence.

Chapter 1 Ingredients of Change: Functions and Models.

Chapter 5 Accumulating Change: Limits of Sums and the Definite Integral.

9-6 RATIO AND ROOT TESTS Miss Battaglia Objective: Test for convergence/ divergence using the ratio and root tests.

Final Exam Term121Term112 Improper Integral and Ch10 16 Others 12 Term121Term112 Others (Techniques of Integrations) 88 Others-Others 44 Remark: ( 24 )

Chapter Five Integration.

Lecture 17 – Sequences A list of numbers following a certain pattern

Copyright © Cengage Learning. All rights reserved.

Chapter 12: Infinite Series

Sequences and Infinite Series

Infinite Sequences and Series

MTH 253 Calculus (Other Topics)

© 2010 Pearson Education, Inc. All rights reserved

Limits and Their Properties

Ratio Test THE RATIO AND ROOT TESTS Series Tests Test for Divergence

For the geometric series below, what is the limit as n →∞ of the ratio of the n + 1 term to the n term?

Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved.

Warm Up Chapter 8.10 Taylor and Maclaurin Series

Copyright © 2006 Pearson Education, Inc

Sequence and Series Dr. Geetha Sivaraman Department of Mathematics

Geometric Sequences and Series

Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved.

Limits and Their Properties

Presentation transcript:

Chapter 9 Infinite Series

Definition of the Limit of a Sequence and Figure 9.1 Copyright © Houghton Mifflin Company. All rights reserved.

Theorem 9.1 Limit of a Sequence Copyright © Houghton Mifflin Company. All rights reserved.

Theorem 9.2 Properties of Limits of Sequences Copyright © Houghton Mifflin Company. All rights reserved.

Theorem 9.3 Squeeze Theorem for Sequences Copyright © Houghton Mifflin Company. All rights reserved.

Theorem 9.4 Absolute Value Theorem Copyright © Houghton Mifflin Company. All rights reserved.

Definition of a Monotonic Sequence Copyright © Houghton Mifflin Company. All rights reserved.

Definition of a Bounded Sequence Copyright © Houghton Mifflin Company. All rights reserved.

Theorem 9.5 Bounded Monotonic Sequences Copyright © Houghton Mifflin Company. All rights reserved.

Definitions of Convergent and Divergent Series Copyright © Houghton Mifflin Company. All rights reserved.

Theorem 9.6 Convergence of a Geometric Series Copyright © Houghton Mifflin Company. All rights reserved.

Theorem 9.7 Properties of Infinite Series Copyright © Houghton Mifflin Company. All rights reserved.

Theorem 9.8 Limit of nth Term of a Convergent Series Copyright © Houghton Mifflin Company. All rights reserved.

Theorem 9.9 nth-Term Test for Divergence Copyright © Houghton Mifflin Company. All rights reserved.

Theorem 9.10 The Integral Test Copyright © Houghton Mifflin Company. All rights reserved.

Theorem 9.11 Convergence of p-Series Copyright © Houghton Mifflin Company. All rights reserved.

Theorem 9.12 Direct Comparison Test Copyright © Houghton Mifflin Company. All rights reserved.

Theorem 9.13 Limit Comparison Test Copyright © Houghton Mifflin Company. All rights reserved.

Theorem 9.14 Alternating Series Test Copyright © Houghton Mifflin Company. All rights reserved.

Theorem 9.15 Alternating Series Remainder Copyright © Houghton Mifflin Company. All rights reserved.

Theorem 9.16 Absolute Convergence Copyright © Houghton Mifflin Company. All rights reserved.

Definitions of Absolute and Conditional Convergence Copyright © Houghton Mifflin Company. All rights reserved.

Theorem 9.17 Ratio Test Copyright © Houghton Mifflin Company. All rights reserved.

Theorem 9.18 Root Test Copyright © Houghton Mifflin Company. All rights reserved.

Guidelines for Testing a Series for Convergence or Divergence Copyright © Houghton Mifflin Company. All rights reserved.

Summary of Tests for Series Copyright © Houghton Mifflin Company. All rights reserved.

Summary of Tests for Series (cont’d) Copyright © Houghton Mifflin Company. All rights reserved.

Definitions of nth Taylor Polynomial and nth Maclaurin Polynomial Copyright © Houghton Mifflin Company. All rights reserved.

Theorem 9.19 Taylor's Theorem Copyright © Houghton Mifflin Company. All rights reserved.

Definition of Power Series Copyright © Houghton Mifflin Company. All rights reserved.

Theorem 9.20 Convergence of a Power Series Copyright © Houghton Mifflin Company. All rights reserved.

Theorem 9.21 Properties of Functions Defined by Power Series Copyright © Houghton Mifflin Company. All rights reserved.

Operations with Power Series Copyright © Houghton Mifflin Company. All rights reserved.

Theorem 9.22 The Form of a Convergent Power Series Copyright © Houghton Mifflin Company. All rights reserved.

Definitions of Taylor and Maclaurin Series Copyright © Houghton Mifflin Company. All rights reserved.

Theorem 9.23 Convergence of Taylor Series Copyright © Houghton Mifflin Company. All rights reserved.

Guidelines for Finding a Taylor Series Copyright © Houghton Mifflin Company. All rights reserved.

Power Series for Elementary Functions Copyright © Houghton Mifflin Company. All rights reserved.