Chapter 7: Q. 35 (page 640)

In Exercises 35–40 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.
$\sum_{k = 1}^{\infty} \frac{5^{k}}{k^{5}}$

Short Answer

Expert verified

The given series is diverges.

Step by step solution

Step 1. Given Information.

The given series is $\sum_{k = 1}^{\infty} \frac{5^{k}}{k^{5}} .$

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,

$ρ = \lim_{k \to \infty} |\frac{c_{k + 1}}{c_{k}}| ρ = \lim_{k \to \infty} |\frac{\frac{5^{k + 1}}{{(k + 1)}^{5}}}{\frac{5^{k}}{k^{5}}}| ρ = \lim_{k \to \infty} |\frac{5^{k + 1} (k^{5})}{5^{k} {(k + 1)}^{5}}| ρ = \lim_{k \to \infty} |\frac{5 k^{5}}{{(k + 1)}^{5}}| ρ = 5 \lim_{k \to \infty} |{(\frac{k}{k + 1})}^{5}| ρ = 5 \lim_{k \to \infty} |{(\frac{1}{1 + \frac{1}{k}})}^{5}| ρ = 5 {(\frac{1}{1 + \frac{1}{\infty}})}^{5} ρ = 5 {(\frac{1}{1 + 0})}^{5} ρ = 5$

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

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In Exercises 35–40 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.
$\sum_{k = 1}^{\infty} \frac{5^{k}}{k^{5}}$

Short Answer

Step by step solution

Step 1. Given Information.

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

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In Exercises 35–40 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.∑k=1∞5kk5

Short Answer

Step by step solution

Step 1. Given Information.

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Applied Mathematics

Pure Maths

Mechanics Maths

Probability and Statistics

Statistics

Study anywhere. Anytime. Across all devices.

In Exercises 35–40 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.
$\sum_{k = 1}^{\infty} \frac{5^{k}}{k^{5}}$