Learning Outcomes
- Determine if a geometric series converges, and if so, the value it converges to.
Section 8.4 Geometric Series (SQ4)
Subsection 8.4.1 Activities
Activity 8.4.1.
Recall from Section 8.3 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 8.3.12, 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 8.4.2.
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 8.4.3.
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 8.4.4.
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 8.4.5.
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 8.4.6.
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 8.4.7.
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 8.4.2 Videos
Subsection 8.4.3 Exercises
Exercises available at
https://tbil.org/calculus/preview/exercises/#/bank/SQ4/
