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Unformatted text preview: ∞ when lim n→∞ n→∞ an
an+1 an
exists,
an+1 = ∞. 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 n→∞ 1 < lim an+1 xn+1
an xn = x n→∞ n→∞ lim an
an+1 = x
,
R lim an
an+1 = x
.
R while it diverges when n→∞ n an xn converges Thus n an xn converges absolutely for x < R, i.e., on the interval (−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 = ∞, the series converges absolutely everywhere on (−∞, ∞). But what happens at x = R if R < ∞ because the Ratio test is inclusive when x = R?
All this is conveniently expressed graphically in Diverges Diverges Converges absolutely −R 0 R Possibl...
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 Spring '11
 Gilbert
 Power Series

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