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Unformatted text preview: ch on letting x → 1− becomes
ln ln(∞) = 1 + 1
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+ + + ... + + ... = ∞.
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n This provides a nice way of actually linking Geometric series and the divergence of the Harmonic series
that wasn’t at all obvious when we started. There must be a bigger picture  that’s what the ﬁnal three
lectures on series are about!!
When an inﬁnite series consists of a sum of multiples of powers of x we make the following Deﬁnition: a series
∞ an xn = a0 + a1 x + a2 x2 + a3 x3 + . . . + an xn + . . .
n=0 is called a Power Series (centered at x = 0).
The coeﬃcients an in a power series totally control what’s going on, and prompt the following Deﬁnition: for a power series n an xn the Radius of Con vergence R is deﬁned by
an
an+1 • R = lim • R = ∞ when lim n...
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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 at Austin.
 Spring '11
 Gilbert
 Power Series

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