# Thinking of the common ratio as a variable x not as a

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Unformatted text preview: monic series in this context. Thinking of the common ratio as a variable x, not as a number r , we get ∞ xn = 1 + x + x2 + x3 + x4 + . . . + xn + . . . = f (x) = n=0 1 1−x (take a = 1 for simplicity). The series converges absolutely on (−1, 1) and on this interval its sum is 1 , so the series provides a series representation of a simple function from high school days. 1−x But this is calculus, so why not integrate both the Geometric series and its sum: x 0 x ∞ ∞ tn dt = n=0 n=0 0 ∞ tn dt = n=0 xn+1 = n+1 ∞ n=0 xn n 1 1 , dt = − ln(1 − x) = ln 1−x 0 1−t assuming that the integral of the inﬁnite sum is the sum of the inﬁnite number of integrals ; that’s x coming up too! Thus 1 x2 x3 x4 xn = x+ + + + ... + + ... , 1−x 2 3 4 n whi...
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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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