Chapter 7: Q. 30 (page 639)

In Exercises 29–34 use the ratio test to analyze whether the given series converges or diverges. If the ratio 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 series diverges.

Step by step solution

Step 1. Given information.

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

Step 2. Ratio test.

According to the given series,

$\frac{a_{k + 1}}{a_{k}} = \frac{\frac{5^{k + 1}}{(k + 1)^{5}}}{\frac{5^{k}}{(k)^{5}}} = \frac{(k)^{5} 5^{k + 1}}{(k + 1)^{5} 5^{k}} = \frac{5 {(k)}^{5}}{(k + 1)^{5}}$

Step 3. Conclusion.

On taking limits,

$\lim_{k \to \infty} \frac{a_{k + 1}}{a_{k}} = \lim_{k \to \infty} \frac{5 (k)^{5}}{(k + 1)^{5}} = 5 (\lim_{k \to \infty} \frac{(k)^{5}}{(k + 1)^{5}}) = 5 (\lim_{k \to \infty} \frac{(k)^{5}}{(k)^{5} {(1 + \frac{1}{k})}^{5}}) = 5 \times 1 = 5 Since, L > 1,$

Therefore, the series diverges.

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In Exercises 29–34 use the ratio test to analyze whether the given series converges or diverges. If the ratio 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. Ratio test.

Step 3. Conclusion.

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In Exercises 29–34 use the ratio test to analyze whether the given series converges or diverges. If the ratio 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. Ratio test.

Step 3. Conclusion.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Calculus

Theoretical and Mathematical Physics

Decision Maths

Statistics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

In Exercises 29–34 use the ratio test to analyze whether the given series converges or diverges. If the ratio test is inconclusive, use a different test to analyze the series.
$\sum_{k = 1}^{\infty} \frac{5^{k}}{k^{5}}$ .