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Unformatted text preview: a ﬁnite or countably inﬁnite set of open
intervals, so F should contain all open and all closed subsets of Ω. Thus, F must contain any set
that is the intersection of countably many open sets, and so on.
The next two examples make use of the formula for the sum of a geometric series, so we derive
it here. A geometric series with ﬁrst term one has the form 1, x, x2 , x3 , · · · . Equivalently, the k th
term is xk for k ≥ 0. Observe that for any value of x :
(1 − x)(1 + x + x2 + · · · + xn ) = 1 − xn+1 ,
because when the product on the left hand side is expanded, the terms of the form xk for 1 ≤ k ≤ n
cancel out. Therefore, for x = 1, the following equivalent expressions hold for the partial sums of
a geometric series:
1 + x + x2 + · · · + xn = n 1 − xn+1
1−x xk = or
k=0 1 − xn+1
.
1−x (1.3) The sum of an inﬁnite series is equal to the limit of the nth partial sum as n → ∞. If x < 1 then
limn→∞ xn+1 = 0. So letting n → ∞ in (1.3) yields that for x < 1 :
1
1...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.
 Spring '08
 Zahrn
 The Land

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