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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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 Winter '08
 JaberAbdualrahman
 Math, Addition, Geometric Series, Summation

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