Preview Activity 8.6.1.
Consider the alternating geometric series
\begin{equation*}
S = \sum_{k=0}^{\infty} (-1)^k \left( \frac{4}{5} \right)^k = 1 - \frac{4}{5} + \frac{16}{25} - \cdots + (-1)^{n-1} \left( \frac{4}{5} \right)^{n-1} + \cdots\text{.}
\end{equation*}
We want to explore how the partial sums of the series compare to and approximate the exact sum of the series, which is
\begin{equation*}
S = \frac{a}{1-r} = \frac{1}{1-\left(\frac{-4}{5}\right)} = \frac{1}{\frac{9}{5}} = \frac{5}{9}\text{.}
\end{equation*}
(a)
Recall that the \(n\)th partial sum, \(S_n\text{,}\) is the sum of the first \(n\) terms of the infinite geometric series \(S\text{.}\) This means that
\begin{equation*}
S_n = \sum_{k=0}^{n-1} (-1)^k \left( \frac{4}{5} \right)^k = 1 - \frac{4}{5} + \frac{16}{25} - \cdots + (-1)^{n-1} \left( \frac{4}{5} \right)^{n-1}\text{.}
\end{equation*}
Note that the exact fractional values of \(S_1\text{,}\) \(S_2\text{,}\) \(\ldots\text{,}\) \(S_6\text{,}\) have been recorded below along with their decimal representations. Your task is to use this information to do some related computations that help us understand the behavior of the series and its partial sums.
First, by computing the differences between \(S_n\) and \(S\) for several different values of \(n\) (recalling that \(S = \frac{5}{9} = 0.\overline{5}\)), fill in the first column of blank spaces provided below. Then, by viewing the series \(S = 1 - \frac{4}{5} + \frac{16}{25} - \cdots + (-1)^{n-1} \left( \frac{4}{5} \right)^{n-1} + \cdots\) as being in the form \(S = a_0 + a_1 + a_2 + \cdots\text{,}\) fill in the last column of blank spaces.
\begin{align*}
n \amp= 1 \amp S_1 \amp= 1 \amp S_1 - S \amp= \fillinmath{XXXX} \amp a_1 \amp= -0.8 \\
n \amp= 2 \amp S_2 \amp= \frac{1}{5} = 0.2 \amp S_2 - S \amp= \fillinmath{XXXX} \amp a_2 \amp= 0.64 \\
n \amp= 3 \amp S_3 \amp= \frac{21}{25} = 0.84 \amp S_3 - S \amp= \fillinmath{XXXX} \amp a_3 \amp= \fillinmath{XXXX} \\
n \amp= 4 \amp S_4 \amp= \frac{41}{125} = 0.328 \amp S_4 - S \amp= \fillinmath{XXXX} \amp a_4 \amp= \fillinmath{XXXX} \\
n \amp= 5 \amp S_5 \amp= \frac{461}{625} = 0.7376 \amp S_5 - S \amp= \fillinmath{XXXX} \amp a_5 \amp= \fillinmath{XXXX} \\
n \amp= 6 \amp S_6 \amp= \frac{1281}{3125} = 0.40992 \amp S_6 - S \amp= \fillinmath{XXXX} \amp a_6 \amp= \fillinmath{XXXX}
\end{align*}
(b)
What do you notice about the differences between \(S_n\) and \(S\) as the value of \(n\) increases? There are at least two important things you can say.
(c)
What do you notice about how the differences between \(S_n\) and \(S\) compare to the value of \(a_n\text{?}\) There are at least two important things you can say.
