Exercise 4 Page 435

We already know the rule for the sum of the first $n$ terms of a geometric series with first term $a_1$ and common ratio $r\ne 1.$ We want to write a rule for the sum of an infinite geometric series. To do so, we will show how the value of $r^n$ changes as $n$ increases. We will examine it in two cases. Let's do it!

Case I

Consider the function $f(n)=r^n,$ where $|r|<1.$ This function represents an exponential decay function. Therefore, it has an horizontal asymptote at the $x\text{-}$axis, or $y=0.$

In this case where the absolute value of $r$ is less than $1,$ this means $r^n$ approaches $0$ as $n$ increases. Since $r^n$ approaches $0$ as $n$ increases, we can ignore $r^n$ in the rule for $S_n$ for large of values of $n.$ Therefore, if the absolute value of the common ratio of an infinite geometric series is less than $1,$ its sum $S$ can be calculated as follows.

Case II

Be aware that when the absolute value of $r$ is greater than $1,$ $r^n$ grows rapidly. With this in mind, let's consider the function $f(n)=r^n,$ where $|r|>1.$

As we can see, the value of $r^n$ becomes extremely large as $n$ increases. Therefore, an infinite geometric series with $|r|>1$ does not have a finite sum.