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:
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  , 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?
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