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Section 8.4 Geometric Series (SQ4)

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, given each restriction on \(r\text{,}\) determine if \(\displaystyle \sum_{n=0}^\infty ar^n\) converges.

  1. \(r>1\text{.}\)

  2. \(r=1\text{.}\)

  3. \(-1<r<1\text{.}\)

  4. \(r=-1\text{.}\)

  5. \(r<-1\text{.}\)

(b)

Where possible, determine what value \(\displaystyle \sum_{n=0}^\infty ar^n\) converges to.

Activity 8.4.3.

Consider the infinite series

\begin{equation*} 5+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots. \end{equation*}

(a)

What of the following approaches will best help us determine the convergence of this series?

  1. This is a geometric series of the form \(\displaystyle \sum_{n=0}^\infty 5\cdot \left(\frac{1}{2}\right)^n\text{.}\)

  2. This is a geometric series of the form \(\displaystyle \sum_{n=0}^\infty 1\cdot \left(\frac{1}{2}\right)^n\text{.}\)

  3. We can rewrite

    \begin{equation*} 5+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots = 4+\left(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots \right). \end{equation*}

(b)

Using your chosen approach, determine if \(5+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots\) converges, and if so, to what it converges.

Activity 8.4.4.

Consider the infinite series

\begin{equation*} \displaystyle\sum_{n=3}^\infty 2(0.7)^n. \end{equation*}

(a)

What of the following approaches will best help us determine the convergence of this series?

  1. Noticing that \(\displaystyle 3\cdot\sum_{n=3}^\infty 2(0.7)^n=\sum_{n=0}^\infty 2(0.7)^n\text{.}\)

  2. Noticing that \(\displaystyle 2+\sum_{n=3}^\infty 2(0.7)^n=\sum_{n=0}^\infty 2(0.7)^n\text{.}\)

  3. Noticing that \(\displaystyle 2n+\sum_{n=3}^\infty 2(0.7)^n=\sum_{n=0}^\infty 2(0.7)^n\text{.}\)

  4. Noticing that \(\displaystyle 2(0.7)^0+2(0.7)^1+2(0.7)^2+\sum_{n=3}^\infty 2(0.7)^n=\sum_{n=0}^\infty 2(0.7)^n\text{.}\)

  5. Noticing that \(\displaystyle 0+1+2+\sum_{n=3}^\infty 2(0.7)^n=\sum_{n=0}^\infty 2(0.7)^n\text{.}\)

(b)

Using your chosen approach, determine if \(\displaystyle\sum_{n=3}^\infty 2(0.7)^n\) converges, and if so, to what it converges.

Example 8.4.5.

Given a series that appears to be mostly geometric:

\begin{equation*} 3+(1.1)^3+(1.1)^4+\cdots(1.1)^n+\cdots \end{equation*}

we can rewrite it as the sum of a standard geometric series with some modification: \(\displaystyle ???+\sum_{n=0}^\infty a\cdot r^n\text{:}\)

\begin{align*} &3+(1.1)^3+(1.1)^4+\cdots(1.1)^n+\cdots \\ &=3-[(1.1)^0+(1.1)^1+(1.1)^2)]+[(1.1)^0+(1.1)^1+(1.1)^2]+(1.1)^3+\cdots\\ &=3-[(1.1)^0+(1.1)^1+(1.1)^2]+(1.1)^0+(1.1)^1+(1.1)^2+(1.1)^3+\cdots\\ &=3-((1.1)^0+(1.1)^1+(1.1)^2)+\sum_{n=0}^\infty (1.1)^n\\ &=-0.31+\sum_{n=0}^\infty (1.1)^n \end{align*}

which converges if and only if \(\displaystyle \sum_{n=0}^\infty (1.1)^n\) converges. In this case, it does not.

Activity 8.4.6.

For each of the following modified geometric series, rewrite them in the form \(???+\displaystyle\sum_{n=0}^\infty a\cdot r^n\text{.}\)

  1. \(\displaystyle -7+\left( -\frac{3}{7}\right)^2+\left( -\frac{3}{7}\right)^3+\cdots\text{.}\)

  2. \(-6+\left(\frac{5}{4}\right)^3+\left(\frac{5}{4}\right)^4+\cdots\text{.}\)

  3. \(\displaystyle 4+\sum_{n=4}^\infty \left(\frac{2}{3}\right)^n\text{.}\)

  4. \(8-1+1-1+1-1+\cdots\text{.}\)

Activity 8.4.7.

Use your rewritten forms from Activity 8.4.6 to determine which of the modified geometric series converge. If the series converges, find to what value it converges.

  1. \(\displaystyle -7+\left( -\frac{3}{7}\right)^2+\left( -\frac{3}{7}\right)^3+\cdots\text{.}\)

  2. \(-6+\left(\frac{5}{4}\right)^3+\left(\frac{5}{4}\right)^4+\cdots\text{.}\)

  3. \(\displaystyle 4+\sum_{n=4}^\infty \left(\frac{2}{3}\right)^n\text{.}\)

  4. \(8-1+1-1+1-1+\cdots\text{.}\)

Activity 8.4.8.

Find the limit of the following series.

\begin{equation*} -1 + \sum_{n = 1 }^\infty 2\cdot\left(\frac{1}{2}\right)^n. \end{equation*}

Subsection 8.4.1 Videos

Figure 152. Video: Determine if a geometric series converges, and if so, the value it converges to.