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:

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Let's multiply both sides by 1/2

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and subtract the second line from the first. All terms on the right side except for the 1 will cancel out! Bingo:

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We have shown that

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One also says that this series converges to 2.


Let's play the same game for a general q instead of 1/2:

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multiply both sides by q

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then, subtract the second line from the first:

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The series

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is called the geometric series. It is the most important series you will encounter!


Example:

Find the sum of the series

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First, factor out the 5 from upstairs and a 2 from downstairs:

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The series in the parentheses is the geometric series with tex2html_wrap_inline218 , but the first term, the "1" at the beginning is omitted. Thus, the series sums up to

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N.B. There is a slightly slicker way to do this. Do you see how?


Try it yourself!

Find the sum of the series

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Click here for the answer or to continue.


Helmut Knaust
Tue Jul 9 16:53:53 MDT 1996

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