Learning Outcomes
- Determine if a geometric series converges, and if so, the value it converges to.
Section 16.4 Geometric Series (SQ4)
Subsection 16.4.1 Activities
Activity 16.57.
Recall from Section 16.3 that for any real numbers \(a, r\) and \(\displaystyle S_n=\sum_{i=0}^n ar^i\) that:
\begin{align*}
S_n=\sum_{i=0}^n ar^i &= a+ar+ar^2+\cdots ar^n\\
(1-r)S_n=(1-r)\sum_{i=0}^n ar^i&= (1-r)(a+ar+ar^2+\cdots ar^n)\\
(1-r)S_n=(1-r)\sum_{i=0}^n ar^i&= a-ar^{n+1}\\
S_n&=a\frac{1-r^{n+1}}{1-r}.
\end{align*}
(a)
Using Definition 16.46, for which values of \(r\) does \(\displaystyle \sum_{n=0}^\infty ar^n\) converges?
- \(|r|>1\text{.}\)
- \(|r|=1\text{.}\)
- \(|r|<1\text{.}\)
- The series converges for every value of \(r\text{.}\)
(b)
Where possible, determine what value \(\displaystyle \sum_{n=0}^\infty ar^n\) converges to.
Fact 16.58.
Geometric series are sums of the form
\begin{equation*}
\sum_{n=0}^\infty ar^n=a+ar+ar^2+ar^3+\dots\text{,}
\end{equation*}
where \(a\) and \(r\) are real numbers. When \(|r|<1\) this series converges to the value \(\displaystyle\frac{a}{1-r}\text{.}\) Otherwise, the geometric series diverges.
Activity 16.59.
Consider the infinite series
\begin{equation*}
5+\frac{3}{2}+\frac{3}{4}+\frac{3}{8}+\cdots.
\end{equation*}
(a)
Complete the following rearrangement of terms.
\begin{align*}
5+\frac{3}{2}+\frac{3}{4}+\frac{3}{8}+\cdots & = \unknown + \left(3+\frac{3}{2}+\frac{3}{4}+\frac{3}{8}+\cdots\right)\\
& = \unknown + \sum_{n=0}^\infty \unknown \cdot \left(\frac{1}{\unknown}\right)^n
\end{align*}
(b)
Since \(|\frac{1}{\unknown}|<1\text{,}\) this series converges. Use the formula \(\sum_{n=0}^\infty ar^n=\frac{a}{1-r}\) to find the value of this series.
- \(\displaystyle \frac{7}{2}\)
- \(\displaystyle \frac{13}{2}\)
- \(\displaystyle 8\)
- \(\displaystyle 10\)
Activity 16.60.
Complete the following calculation, noting \(|0.6|<1\text{:}\)
\begin{align*}
\sum_{n=2}^\infty 2(0.6)^n &=\left(\sum_{n=0}^\infty 2(0.6)^n\right) - \unknown - \unknown \\
& = \left(\frac{\unknown}{1-\unknown}\right)- \unknown - \unknown
\end{align*}
What does this simplify to?
- \(\displaystyle 1.1\)
- \(\displaystyle 1.4\)
- \(\displaystyle 1.8\)
- \(\displaystyle 2.1\)
Observation 16.61.
Given a series that appears to be mostly geometric such as
\begin{equation*}
3+(1.1)^3+(1.1)^4+\cdots(1.1)^n+\cdots
\end{equation*}
we can always rewrite it as the sum of a standard geometric series with some finite modification, in this case:
\begin{equation*}
-0.31 + \sum_{n=0}^\infty (1.1)^n
\end{equation*}
Thus the original series converges if and only if \(\displaystyle \sum_{n=0}^\infty (1.1)^n\) converges.
When the series diverges as in this example, then the reason why (\(|1.1|\geq 1\)) can be seen without any modification of the original series.
Activity 16.62.
For each of the following modified geometric series, determine without rewriting if they converge or diverge.
(a)
\(\displaystyle -7+\left( -\frac{3}{7}\right)^2+\left( -\frac{3}{7}\right)^3+\cdots\text{.}\)
(b)
\(-6+\left(\frac{5}{4}\right)^3+\left(\frac{5}{4}\right)^4+\cdots\text{.}\)
(c)
\(\displaystyle 4+\sum_{n=4}^\infty \left(\frac{2}{3}\right)^n\text{.}\)
(d)
\(8-1+1-1+1-1+\cdots\text{.}\)
Activity 16.63.
Find the value of each of the following convergent series.
(a)
\(-1 + \sum_{n = 1 }^\infty 2\cdot\left(\frac{1}{2}\right)^n\text{.}\)
(b)
\(\displaystyle -7+\left( -\frac{3}{7}\right)^2+\left( -\frac{3}{7}\right)^3+\cdots\text{.}\)
(c)
\(\displaystyle 4+\sum_{n=4}^\infty \left(\frac{2}{3}\right)^n\text{.}\)
Subsection 16.4.2 Videos
Subsection 16.4.3 Exercises
Exercises available at
https://tbil.org/preview/calculus/exercises/#/bank/SQ4/
