Exercise 7

In Equation, we learned that for an infinite geometric series $$\begin{gathered} S = a + ar + ar^2 + \cdots + ar^{n-1} + \cdots \end{gathered}$$, if $|r| \lt 1$, then $$\begin{gathered} S = \frac{a}{1-r} \end{gathered}$$. This result provides a quick way to determine the value of an infinite geometric series that converges.

In this exercise, we explore why the infinite geometric series fails to converge when $r = 1$, $r = -1$, and $|r|>1$.

1. If we consider the finite geometric series that results when $a = 1$ and $r = 1$, we get the sum $$\begin{gathered} S_n = 1 + 1 + \cdots + 1 \end{gathered}$$, where there are $n$ terms in the sum.

   1. Compute $S_2$, $S_3$, and $S_4$. What is the general formula for $S_n$?

   2. Explain why the sequence $S_n$ does not converge to a finite value. Contrast this with, for example, our work in Example.

2. Next, consider the finite geometric series that results when $a = 1$ and $r = -1$, which is the sum $$\begin{gathered} S_n = 1 - 1 + 1 - 1 + \cdots + (-1)^{n-1} \end{gathered}$$, where there are $n$ terms in the sum. (We view the first “$1$” as resulting from “$1 \cdot (-1)^0$”.)

   1. Compute $S_2$, $S_3$, $S_4$, and $S_5$. What do you observe?

   2. Explain why the sequence $S_n$ does not converge to a finite value. Contrast this with, for example, our work in Example.

3. Now consider the geometric sum with $a = 1$ and $r = 2$, so $$\begin{gathered} S_n = 1 + 2 + 4 + \cdots + 2^{n-1} \end{gathered}$$.

   1. Compute $S_2$, $S_3$, and $S_4$. What is the general formula for $S_n$?

   2. What do you observe happens to these partial sums as $n$ increases without bound? What does this tell us about the infinite geometric series $1 + 2 + 4 + \cdots + 2^{n-1} + \cdots$?