Chapter 7: Q. 13 (page 656)

Geometric series: For each of the series that follow, find the sum or explain why the series diverges.
$\sum_{k = 4}^{\infty} {(\frac{1}{3})}^{k + 2}$

Short Answer

Expert verified

The sum of the series is $\begin{array}{rcl} \frac{1}{2} {(\frac{1}{3})}^{5} \end{array}$ .

Step by step solution

Step 1. Given Information

The given series is $\sum_{k = 4}^{\infty} {(\frac{1}{3})}^{k + 2}$ .

Step 2. Determine if a geometric series

Find the ratio of the consecutive terms.

$\begin{array}{rcl} \frac{{(\frac{1}{3})}^{k + 2 + 1}}{{(\frac{1}{3})}^{k + 2}} & = & {(\frac{1}{3})}^{(k + 2 + 1) - (k + 2)} \\ = & {(\frac{1}{3})}^{1} \\ = & \frac{1}{3} \end{array}$

Since the ratio of the consecutive terms is same, the given series is a geometric series with a common ratiorole="math" localid="1649256775632" $r = \frac{1}{3}$ .

Step 3. Determine the convergence

Determine the first term and the common ratio.

$\begin{array}{rcl} c & = & {(\frac{1}{3})}^{4 + 2} \\ = & {(\frac{1}{3})}^{6} \\ r & = & \frac{1}{3} \end{array}$

If $|r| < 1$ , the series converges to $\frac{c}{1 - r}$ .

$\begin{array}{rcl} \frac{c}{1 - r} & = & \frac{{(\frac{1}{3})}^{6}}{1 - (\frac{1}{3})} \\ = & \frac{{(\frac{1}{3})}^{6}}{(\frac{2}{3})} \\ = & \frac{1}{2} {(\frac{1}{3})}^{5} \end{array}$

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Geometric series: For each of the series that follow, find the sum or explain why the series diverges.
$\sum_{k = 4}^{\infty} {(\frac{1}{3})}^{k + 2}$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Determine if a geometric series

Step 3. Determine the convergence

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Geometric series: For each of the series that follow, find the sum or explain why the series diverges.∑k=4∞13k+2

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Determine if a geometric series

Step 3. Determine the convergence

One App. One Place for Learning.

Most popular questions from this chapter

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Study anywhere. Anytime. Across all devices.

Geometric series: For each of the series that follow, find the sum or explain why the series diverges.
$\sum_{k = 4}^{\infty} {(\frac{1}{3})}^{k + 2}$