PPT-Goal: Does a series converge or diverge?

Lecture 24 Divergence Test 1 Divergence Test If a series converges then sequence converges to 0 Example 1 ConvergeDiverge 2 Example 2 ConvergeDiverge Example 3 ConvergeDiverge. DEFINITION. Absolute . Convergence. Verify that the series. converges absolutely.. This series converges absolutely because the positive series . (. with . absolute values. ) is a . p. -series with . Alternating Series . A series whose terms are alternately positive and negative. If n starts at 1, then the first . term . is negative. If n starts at 1, then the first term is positive. Alternating Series Test . Consider and. 1. Show that the Ratio Test yields for both series.. For :. For :. Do Now: Exploration 1 on p.492. Consider and. 2. Use improper integrals to find the areas shaded in Figures. arise in applications, but the convergence tests developed so far cannot be applied easily. Fortunately, the Ratio Test can be used for this and many other series.. THEOREM . 1 . Ratio . Test. . Assume . 9-D . Comparison Tests . The Direct comparison Test . If 0 < a. n . < . b. n. . . for all n and positive terms, then:. If the larger series converges, then the smaller series must also converge. . Sum of a Geometric Series. Let . c. 0. . If |. r. | < 1, then. If |. r. | ≥ 1, then the geometric series diverges.. Sum of an Infinite Geometric Series (80). Sum of an Infinite Geometric Series (80). Convergence . Tests and Taylor . Series. Part I: Convergence Tests. Objectives. Know how to decide if a series converges or not. Corresponding sections in Simmons: 13.5, 13.6,13.7,13.8. Important examples. (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. 4, … . a. n. …. {(1, 1), (2, ½), (3, ¼), (4, 1/8) ….}. {. a. n. } = . a. 1 . , a. 2 . , a. 3 . , a. 4 . , … , a. n . , . …. Pattern is determined by position or by what has come before. 3, . 6. , . 12. , . . 24, . 48. , . …. Absolute Convergence . Conditional Convergence . 6. Does . . converge or diverge?. If it converges, is it conditionally or . absolutely convergent?. 7. Does . . converge or diverge?. If . Occasionally, a series may have both positive and negative terms and not be an alternating series. For instance, the series. has both positive and negative terms, yet it is not an alternating series. One way to obtain some information about the convergence of this series is to investigate the convergence of the series.  . Section 8.6 – Trapezoid Rule. Used to estimate the area under a curve .  . Example:. Estimate the area under the curve of . from . to . using the trapezoid rule with n = 4..  . 8.7 Improper Integrals. . Revised . 24 October 09. M. . Pasenelli. CACPFO Rules Interpreter. 2. Our Mission. Review the rules and mechanics for overtime. Demonstrate our knowledge of the rules by answering the quiz questions correctly. © 2012 Rules No calculators, books, or phones. Teams alternate choosing targets. Every person takes a turn and can consult team members before answering the question. If correct, the target is revealed. The first team to uncover all 3 battleships wins.