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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.
 Spring '11
 Gilbert
 Power Series

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