# PP 12.8_1 - Power Series Definition where x is a variable...

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Unformatted text preview: Power Series Definition where x is a variable and the c n ’s are constants called the coefficients of the series. 2 3 1 2 3 ... n n n c x c c x c x c x ∞ = = + + + + ∑ A power series is a series of the form whose domain is the set of all x for which the series converges. 2 1 2 ( ) ... ... n n f x c c x c x c x = + + + + + The sum of the series is a function Example . series power the of domain the Determine n ∑ ∞ = n x ∑ ∑ ∞ =- ∞ = = 1 1 m m n n x x So, we have a geometric series with r = x . By the Geometric Series Test, the series converges for | r = x| < 1 and diverges otherwise. Since all tests available start the sum at n = 1, we need to modify the given series by making m = n + 1. Hence the domain of the power series is the interval (-1, 1). Notation: Since the domain of a power series is usually an interval of real numbers, it is called the interval of convergence of the power series. Thus, the interval of convergence of the series under consideration is (-1, 1). 2 1 2 ( ) ( ) ( ) ... n n n c x a c c x a c x a ∞ =- = +- +- + ∑ Generalization A series of the form is called a power series centered at a or a power series about a Examples 0. at centered is ∑ ∞ = n n x 1. at centered is ) 1 ( 2 ∑ ∞ =- n n n x . at centered is )! 2 ( ) ( π π- + ∑ ∞ = n n n x Comments The sum of a power series resembles a polynomial....
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PP 12.8_1 - Power Series Definition where x is a variable...

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