The Geometric Series


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:


The series


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


Find the sum of the series


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


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


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

Try it yourself!

Find the sum of the series


Click here for the answer or to continue.

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

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