Well the ratio test says that absolutely when 1 lim

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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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