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2414 Calculus II Chapter 9(2) Power Series Convergence of Power Series

2414 Calculus II Chapter 9(2) Power Series Convergence of Power Series. Power Series. If x is a variable then. is called a Power Series. is a Power Series Centered at the constant c. Radius of Convergence For a Power Series centered at c , only one of the following can happen:.

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2414 Calculus II Chapter 9(2) Power Series Convergence of Power Series

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  1. 2414 Calculus IIChapter 9(2)Power SeriesConvergence of Power Series

  2. Power Series If x is a variable then is called a Power Series is a Power Series Centered at the constant c

  3. Radius of ConvergenceFor a Power Series centered at c, only one of the following can happen: 1) The series converges only at c R = 0 • There is a number R > 0 so that the series • converges absolutely for |x – c| < R and • diverges for |x – c| > R • R is the Radius of Convergence R = a # R = ∞ 3) The series converges absolutely for all x The set of all x that make the series converge is the Interval of Convergence ( ) R R C R=0

  4. Find the Radius of Convergence of: Use the Ratio Test since we have factorials. This Diverges. Divergence eliminates the second two choices so you are back to converging only at c and R = 0

  5. Find the Radius of Convergence of Since we have factorial try the ratio test. Since 0 < 1, this always converges. The Radius of convergence is R = 

  6. Find the Radius and Interval of Convergence for: Three tests can be used: Geometric Series, Root, Ratio Geometric is the “easiest” The Radius of Convergence is R = 1 The Interval of Convergence is 1 unit from “c” or (1,3)

  7. Find the Interval of Convergence of Since we have powers try the ratio test. By Ratio test Converges if 2x < 1. The Radius is R = ½. The interval will be

  8. Find the Interval of Convergence of Since we have a power try the root test. By Root test Converges if x/2 < 1. The Radius is R = 2. The interval will be

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