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# 5.1we - 5.1 REVIEW OF POWER SERIES KIAM HEONG KWA For a...

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5.1: REVIEW OF POWER SERIES KIAM HEONG KWA For a sequence of (real) numbers { a n } n =0 and a fixed (real) number x 0 , a series of the form (1) X n =0 a n ( x - x 0 ) n is called a power series of the variable x centered at x 0 . It defines a function f ( x ) of x whenever it is convergent in the following sense. The power series (1) is said be convergent at x = x 1 if the limit lim N →∞ N X n =0 a n ( x 1 - x 0 ) n exists as a finite number. In this case, we define f ( x 1 ) = X n =0 a n ( x 1 - x 0 ) n = lim N →∞ N X n =0 a n ( x 1 - x 0 ) n . Note that f ( x 0 ) = 0 trivially. If the interval I is the maximal interval on which f ( x ) is defined, i.e., if I is the largest interval in which the power series (1) is convergent, then I is called the interval of convergence of the power series. In other words, we can write (2) f ( x ) = X n =0 a n ( x - x 0 ) n for all x I . It is a consequence of the ratio test, which we shall describe below, that if I is bounded, then it has one of the following forms: (3) ( x 0 - ρ, x 0 + ρ ) , [ x 0 - ρ, x 0 + ρ ) , ( x 0 - ρ, x 0 + ρ ] , or [ x 0 - ρ, x 0 + ρ ] , for some nonnegative constant ρ . Note that ρ is half of the length of the interval I . It is usually known as the radius of convergence of the power series. If I is unbounded, then it must be the whole real line, i.e., I = R . In this case, we write ρ = and say that the power series has an infinite radius of convergence. Date : February 4, 2011. 1

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2 KIAM HEONG KWA Exercise 1. Show that if ρ 1 and ρ 2 are, respectively, radii of conver- gence of the power series n =0 a n ( x - x 0 ) n and n =0 b n ( x - x 0 ) n , then the radius of convergence ρ of X n =0 ( a n + b n )( x - x 0 ) n is given by ρ = min { ρ 1 , ρ 2 } .
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