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Calculus II (MAT 146) Dr. Day Monday April 28, 2014

Calculus II (MAT 146) Dr. Day Monday April 28, 2014. Transforming Series into Functions Power Series Coefficients Applying the Ratio Test to Power Series Radius of Convergence Interval of Convergence Semester Exam Reviews Vizor Center : Tonight! 7-9 pm Vrooman 108

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Calculus II (MAT 146) Dr. Day Monday April 28, 2014

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  1. Calculus II (MAT 146)Dr. Day Monday April 28, 2014 • Transforming Series into Functions • Power Series • Coefficients • Applying the Ratio Test to Power Series • Radius of Convergence • Interval of Convergence • Semester Exam Reviews • Vizor Center: Tonight! 7-9 pm Vrooman 108 • Class Help Session: Sunday, May 4, 6-7 pm, STV 325 Extra-Credit Video Due Tonight at Midnight! • Need completed form in my hand! • Send me an email with a link to your YouTube video! MAT 146

  2. Assignments Homework Tasks • Wednesday: Quiz #10 1. converge/diverge 2. Convergence Interval • This Week: 11.8 (Today), 11.9, 11.10 • By the Day of Semester Exam: WA Review Assignments (3) MAT 146

  3. Power Series • x is a variable. • The cn’s are constants, called the coefficients. • For any fixed value of x, we can test the series for convergence. MAT 146

  4. Power Series • The sum of the series is a function with domain the set of all x values for which the series converges. • The function seems to be a polynomial, except it has an infinite number of terms. MAT 146

  5. Power Series: Example • If we let cn = 1 for all n, we get a familiar series: • This geometric series has common ratio x and we know the series converges for |x| < 1. • We also know the sum of this series: MAT 146

  6. Generalized Power Series • This is called: • a power series in (x – a), or • a power series centered at a, or • a power series about a. MAT 146

  7. Power Series Convergence • For what values of x does this series converge? • Determine its Radius of Convergence and its Interval of Convergence. MAT 146

  8. Power Series Convergence • For what values of x does this series converge? • Determine its Radius of Convergence and its Interval of Convergence. MAT 146

  9. Power Series Convergence • For what values of x does this series converge? • Use the Ratio Test to determine values of x that result in a convergent series. MAT 146

  10. Power Series Convergence • For what values of x does this series converge? • Use the Ratio Test to determine values of x that result in a convergent series. MAT 146

  11. Power Series Convergence • For what values of x does this series converge? • Determine its Radius of Convergence and its Interval of Convergence. MAT 146

  12. Power Series Convergence • For what values of x does this series converge? • Determine its Radius of Convergence and its Interval of Convergence. MAT 146

  13. Geometric Power Series • If we let cn = 1 for all n, we get a familiar series: • This geometric series has common ratio x and we know the series converges for |x| < 1. • We also know the sum of this series: MAT 146

  14. Geometric Power Series MAT 146

  15. Geometric Power Series MAT 146

  16. Geometric Power Series MAT 146

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