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Calculus II (MAT 146) Dr. Day Monday Nov 4, 2013

Calculus II (MAT 146) Dr. Day Monday Nov 4, 2013. Convergent and Divergent Series (11.2) Series Worth Remembering Geometric Series Harmonic Series Telescoping Sums Our First Convergence Test The n th -Term Test aka The Divergence Test Assignments and Announcements.

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Calculus II (MAT 146) Dr. Day Monday Nov 4, 2013

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  1. Calculus II (MAT 146)Dr. Day Monday Nov 4, 2013 • Convergent and Divergent Series (11.2) • Series Worth Remembering • Geometric Series • Harmonic Series • Telescoping Sums • Our First Convergence Test • The nth-Term Test aka The Divergence Test • Assignments and Announcements MAT 146

  2. What is an Infinite Series? We start with a sequence {an}, ngoing from 1 to ∞, and define {si} as shown. The {si} are called partial sums. These partial sums themselves form a sequence. An infinite series is the summation of an infinite number of terms of the sequence {an}. MAT 146

  3. What is an Infinite Series? Our goal is to determine whether an infinite series converges or diverges. It must do one or the other. If the sequence of partial sums {si} has a finite limit as n −−> ∞, we say that the infinite series converges. Otherwise, it diverges. MAT 146

  4. Notable Series A geometric series is created from a sequence whose successive terms have a common ratio. When will a geometric series converge? MAT 146

  5. Notable Series The harmonic series is the sum of all possible unit fractions. MAT 146

  6. Notable Series A telescoping sum can be compressed into just a few terms. MAT 146

  7. Fact or Fiction? MAT 146

  8. A Series Convergence Test:The nth-Term Testalso calledThe Divergence Test MAT 146

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  10. Geometric Series (1) Create a geometric series to represent the repeating decimal fraction 0.45454545454545… and then determine the common fraction equivalent to this repeating decimal fraction. • (2) A geometric series has first term a1 = 1 and common ratio x, |x| < 1. • (a) Represent this using sigma notation and then show the first five terms of the series. • (b) Using your knowledge of convergent geometric series, determine a representation for this sum. MAT 146

  11. Series Convergence or Divergence We have defined sequences and series and have considered a few particular series—geometric series, harmonic series, telescoping sums—and explored the convergence/divergence of these series. Our first test related to this is • The Divergence Test also known as The nth-Term Test: If the sequence of terms that comprise an infinite series DO NOT have a limit of 0, then the series DOES NOT converge. The series diverges. We now turn our attention more fully to the fate of infinite series. What strategies do we have for determining whether an infinite series converges or diverges? MAT 146

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  19. Assignments WebAssign • Ch 11.2 due tonight at midnight MAT 146

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