# 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:

The series

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

#### Example:

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?

#### Try it yourself!

Find the sum of the series