2 3 4 n this provides a nice way of actually

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Unformatted text preview: ch on letting x → 1− becomes ln ln(∞) = 1 + 1 1 1 1 + + + ... + + ... = ∞. 2 3 4 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.

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