Intro to Analysis in-class_Part_56

# Intro to Analysis in-class_Part_56 - 7.4 Power series 117...

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Unformatted text preview: 7.4 Power series 117 7.4 Power series 7.4.1 Radius of convergence Definition 7.4.1. A power series is a series of the form ∑ a n x n , where x is a variable. The n th term of a power series is f n ( x ) = a n x n (rather than just a n ). ∑ a n x n is a family of series, one for each value of x . We are interested in the subfamily corresponding to A = { x ∈ R . . . X | a n x n | converges } . Then we can define a function f : A → R , f ( x ) = X a n x n . Definition 7.4.2. For any power series ∑ a n x n , ∃ ! R ≥ 0 such that X a n x n converges absolutely for | x | < R, X a n x n diverges for | x | > R. R is the radius of convergence of the power series. (Note: may have R = 0.) By convention, R = ∞ iff the series converges ∀ x ∈ R . Now must validate the assertion of the definition : that such an R exists. This will be shown by computing R explicitly. Theorem 7.4.3. The radius R of a power series ∑ c n z n is given by 1 R = limsup n →∞ n p | c n | ....
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Intro to Analysis in-class_Part_56 - 7.4 Power series 117...

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