# Therefore for x 1 the following equivalent

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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| &lt; 1 then limn→∞ xn+1 = 0. So letting n → ∞ in (1.3) yields that for |x| &lt; 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.

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