But y 1 y 1 x 1 b x 1 b x b x

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Unformatted text preview: is a positive constant, the power series x x 1+ + b b 2 3 x + b ∞ + ··· = n=0 x b n (1.2) has radius of convergence b. To see this, we make the substitution y = x/b, ∞ and the power series becomes n=0 y n , which we already know converges for |y | < 1 and diverges for |y | > 1. But |y | < 1 ⇔ |y | > 1 ⇔ x <1 b x >1 b ⇔ |x| < b, ⇔ |x| > b. 1 Good references for the theory behind convergence of power series are Edward D. Gaughan, Introduction to analysis , Brooks/Cole Publishing Company, Pacific Grove, 1998 and Walter Rudin, Principles of mathematical analysis , third edition, McGraw-Hill, New York, 1976. 3 Thus for |x| < b the power series (1.2) converges to 1 1 b = = , 1−y 1 − (x/b) b−x while for |x| > b, it diverges. There is a simple criterion that often enables one to determine the radius of convergence of a power series. Ratio Test. The radius of convergence of the power series ∞ a0 + a1 (x − x0 ) + a2 (x − x0 )2 + · · · = an (x − x0 )n n=0 is given by the formula R = lim n→∞ |an | , |an+1 | so long as t...
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This document was uploaded on 01/12/2014.

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