Skip to main content
Contents Index Dark Mode Prev Up Next \(\newcommand{\dollar}{\$}
\DeclareMathOperator{\erf}{erf}
\DeclareMathOperator{\arctanh}{arctanh}
\DeclareMathOperator{\arcsec}{arcsec}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Activity 8.2.3 .
Let \(r \ne 1\) and \(a\) be real numbers and let
\begin{equation*}
S = a+ar+ar^2 + \cdots ar^{n-1} + \cdots
\end{equation*}
be an infinite geometric series. For each positive integer \(n\text{,}\) let
\begin{equation*}
S_n = a+ar+ar^2 + \cdots + ar^{n-1}\text{.}
\end{equation*}
Recall that
\begin{equation*}
S_n = a\frac{1-r^n}{1-r}\text{.}
\end{equation*}
(a) What should we allow
\(n\) to approach in order to have
\(S_n\) approach
\(S\text{?}\)
(b) What is the value of
\(\lim_{n \to \infty} r^n\) for
\(|r| \gt 1\text{?}\) for
\(|r| \lt 1\text{?}\) Explain.
(c) If
\(|r| \lt 1\text{,}\) use the formula for
\(S_n\) and your observations in (a) and (b) to explain why
\(S\) is finite and find a resulting formula for
\(S\text{.}\)
