Thus for example 1x 1 is analytic whenever x0 1 but

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Unformatted text preview: . In more advanced courses, one studies criteria under which functions are real analytic. For our purposes, it is suﬃcient to be aware of the following facts: The sum and product of real analytic functions is real analytic. It follows from this that any polynomial P (x) = a0 + a1 x + a2 x2 + · · · + an xn is analytic at any x0 . The quotient of two polynomials with no common factors, P (x)/Q(x), is analytic at x0 if and only if x0 is not a zero of the denominator Q(x). Thus for example, 1/(x − 1) is analytic whenever x0 = 1, but fails to be analytic at x0 = 1. Exercises: 1.1.1. Use the ratio test to ﬁnd the radius of convergence of the following power series: ∞ ∞ 1 a. xn , (−1)n xn , b. n+1 n=0 n=0 ∞ ∞ 3 (x − 2)n , n+1 n=0 c. d. ∞ 1 (x − π )n , 2n n=0 ∞ (7x − 14)n , e. f. n=0 1 (3x − 6)n . n! n=0 1.1.2. Use the comparison test to ﬁnd an estimate for the radius of convergence of each of the following power series: ∞ a. k=0 ∞ c. k=0 ∞ 1 2k x, (2k )! (−1)k x2k , b. k=0 ∞ 1 (x − 4)2k 2k d....
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This document was uploaded on 01/12/2014.

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