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Chapter 7: Q. 35 (page 640)

In Exercises 35鈥�40 use the root test to analyze whether the given series converges or diverges. If the root test is inconclusive, use a different test to analyze the series.
${鈭�}_{k = 1}^{鈭�} \frac{5^{k}}{k^{5}}$

Short Answer

Expert verified

The given series is diverges.

Step by step solution

01

Step 1. Given Information.

The given series is ${鈭�}_{k = 1}^{鈭�} \frac{5^{k}}{k^{5}} .$

02

Step 2. Determine whether the given series converges or diverges.

From the series, we can depict that if $c_{k} = \frac{5^{k}}{k^{5}}$ then $c_{k + 1} = \frac{5^{k + 1}}{{(k + 1)}^{5}} .$

So,

$蚁 = \lim_{k 鈫� 鈭�} |\frac{c_{k + 1}}{c_{k}}| 蚁 = \lim_{k 鈫� 鈭�} |\frac{\frac{5^{k + 1}}{{(k + 1)}^{5}}}{\frac{5^{k}}{k^{5}}}| 蚁 = \lim_{k 鈫� 鈭�} |\frac{5^{k + 1} (k^{5})}{5^{k} {(k + 1)}^{5}}| 蚁 = \lim_{k 鈫� 鈭�} |\frac{5 k^{5}}{{(k + 1)}^{5}}| 蚁 = 5 \lim_{k 鈫� 鈭�} |{(\frac{k}{k + 1})}^{5}| 蚁 = 5 \lim_{k 鈫� 鈭�} |{(\frac{1}{1 + \frac{1}{k}})}^{5}| 蚁 = 5 {(\frac{1}{1 + \frac{1}{鈭�}})}^{5} 蚁 = 5 {(\frac{1}{1 + 0})}^{5} 蚁 = 5$

Since $5 > 1$ by using the root test the given series is diverging.

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Most popular questions from this chapter

What is meant by the remainder $R_{n}$ of a series ${鈭�}_{k = 1}^{鈭�} a_{k}$

Determine whether the series ${鈭�}_{k = 0}^{鈭�} \frac{{(- 3)}^{k + 1}}{4^{k - 2}}$ converges or diverges. Give the sum of the convergent series.

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.

${鈭�}_{k = 1}^{鈭�} 鈥� \frac{1}{k^{k}}$

Prove that if ${鈭�}_{k = 1}^{鈭�} a_{k}$ converges to L and ${鈭�}_{k = 1}^{鈭�} b_{k}$ converges to M , then the series ${鈭�}_{k = 1}^{鈭�} (a_{k} + b_{k}) = L + M$ .

Given a series ${鈭�}_{k = 1}^{鈭�} a_{k}$ , in general the divergence test is inconclusive when . For a $a_{k} 鈫� 0$ geometric series, however, if the limit of the terms of the series is zero, the series converges. Explain why.

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