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Chapter 7: Q. 13 (page 656)

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

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

01

Step 1. Given Information

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

02

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" $r = \frac{1}{3}$ .

03

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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Most popular questions from this chapter

Use the divergence test to analyze the given series. Each answer should either be the series diverges or the divergence test fails, along with the reason for your answer.

${鈭�}_{k = 1}^{鈭�} \frac{k}{3 k + 100}$

In Exercises 48鈥�51 find all values of p so that the series converges.

${鈭�}_{k = 1}^{鈭�} \frac{k}{{(1 + k^{2})}^{p}}$

Explain why the integral test may be used to analyze the given series and then use the test to determine whether the series converges or diverges.

${鈭�}_{k = 1}^{鈭�} 鈥� (3 k 鈭� 1)^{鈭� 2 / 3}$

Prove Theorem 7.25. That is, show that the series ${鈭�}_{k = 1}^{鈭�} a_{k} a n d {鈭�}_{k = M}^{鈭�} a_{k}$ either both converge or both diverge. In addition, show that if ${鈭�}_{k = M}^{鈭�} a_{k}$ converges to L, then ${鈭�}_{k = 1}^{鈭�} a_{k}$ converges tolocalid="1652718360109" $a_{1} + a_{2} + a_{3} + . . . . + a_{M - 1} + L .$

Explain why the integral test may be used to analyze the given series and then use the test to determine whether the series converges or diverges.

${鈭�}_{k = 2}^{鈭�} 鈥� \frac{1}{k (\ln 鈦� k)^{2}}$

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