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No Title http://www.sosmath.com/calculus/geoser/geoser01.html 1 of 2 1/31/08 8:47 AM The Geometric Series Introduction Suppose someone offers you the following deal: You get $1 on the first day, $0.50 the second day, $0.25 the third day, and so on. For a second, you might dream about infinite riches, but adding some of the numbers on your calculator will soon convince you that this is an offer for about $2.00, spread out over quite some time. The process of adding infinitely many numbers is at the heart of the mathematical concept of a numerical series . Let's see why the deal above amounts to just $2.00. Let s denote the sum of the series just considered: Let's multiply both sides by 1/2 and subtract the second line from the first. All terms on the right side except for the 1 will cancel out! Bingo: We have shown that One also says that this series converges to 2. Let's play the same game for a general q instead of 1/2: multiply both sides by q then, subtract the second line from the first:
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