Unformatted text preview: †’âˆž when lim nâ†’âˆž nâ†’âˆž an
an+1 an
exists,
an+1 = âˆž. But why is R called the Radius of Convergence? Well, the Ratio test says that
absolutely when:
1 > lim an+1 xn+1
an xn = x nâ†’âˆž 1 < lim an+1 xn+1
an xn = x nâ†’âˆž nâ†’âˆž lim an
an+1 = x
,
R lim an
an+1 = x
.
R while it diverges when nâ†’âˆž n an xn converges Thus n an xn converges absolutely for x < R, i.e., on the interval (âˆ’R, R), and diverges for x > R. So, as measured from where the power series is centered, R is the radius of the interval on
which the series converges absolutely. If R = âˆž, the series converges absolutely everywhere on (âˆ’âˆž, âˆž). But what happens at x = R if R < âˆž because the Ratio test is inclusive when x = R?
All this is conveniently expressed graphically in Diverges Diverges Converges absolutely âˆ’R 0 R Possibl...
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This note was uploaded on 06/05/2011 for the course MATH 408 D taught by Professor Gilbert during the Spring '11 term at University of Texas.
 Spring '11
 Gilbert
 Power Series

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