Math 321 - The Dirichlet Test

Math 321 The Dirichlet Test

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Unformatted text preview: R −1 = z R z n+1 R −1 n ∞ z is independent of n. So the geometric series n=0 R , which has radius of convergence R, converges if and only if |z | < R. ∞ 1 zn The third series, , converges for all |z | ≤ R, by comparison with n=0 n2 R ∞ 1 n=0 n2 . As the series has radius of convergence R, it converges if and only if |z | ≤ R. ∞ 1zn The middle series n=0 n R has a more interesting domain of convergence. Of course the radius of convergence is exactly R, so the series converges for all complex numbers z with |z | < R and diverges for all complex numbers with |z | > R....
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This document was uploaded on 02/24/2014 for the course MATH 321 at University of British Columbia.

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