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Chapter 7: Q. 39 (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.
${鈭�}_{k = 1}^{鈭�} {(\frac{1 + 4 k}{5 k - 1})}^{k}$

Short Answer

Expert verified

The given series converges.

Step by step solution

01

Step 1. Given Information.

The given series is ${鈭�}_{k = 1}^{鈭�} {(\frac{1 + 4 k}{5 k - 1})}^{k} .$

02

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

By using the root test, let the general term is $a_{k} = {(\frac{1 + 4 k}{5 k - 1})}^{k} .$

So,

$蚁 = \lim_{k 鈫� 鈭�} {(a_{k})}^{\frac{1}{k}} 蚁 = \lim_{k 鈫� 鈭�} {({(\frac{1 + 4 k}{5 k - 1})}^{k})}^{\frac{1}{k}} 蚁 = \lim_{k 鈫� 鈭�} (\frac{1 + 4 k}{5 k - 1}) 蚁 = \lim_{k 鈫� 鈭�} (\frac{k (\frac{1}{k} + 4)}{k (5 - \frac{1}{k})}) 蚁 = \frac{4}{5}$

Since $\frac{4}{5} < 1,$ by using the root test the given series converge.

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Most popular questions from this chapter

Let ${鈭�}_{k = 0}^{鈭�} c r^{k}$ and ${鈭�}_{k = 0}^{鈭�} b v^{k}$ be two convergent geometric series. If b and v are both nonzero, prove that ${鈭�}_{k = 0}^{鈭�} \frac{c r^{k}}{b v^{k}}$ is a geometric series. What condition(s) must be met for this series to converge?

Determine whether the series ${鈭�}_{k = 0}^{鈭�} \frac{2^{k + 2}}{5^{k - 1}}$ converges or diverges. Give the sum of the convergent series.

Express each of the repeating decimals in Exercises 71鈥�78 as a geometric series and as the quotient of two integers reduced to lowest terms.

$0.99999 . . .$

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.

${鈭�}_{k = 4}^{鈭�} 鈥� \frac{2 k 鈭� 4}{k^{2} 鈭� 4 k + 3}$

Let ${鈭�}_{k = 1}^{鈭�} a_{k} b e a c o n v e r g e n t s e r i e s a n d {鈭�}_{k = 1}^{鈭�} b_{k} b e a d i v e r g e n t s e r i e s .$ Prove that the series ${鈭�}_{k = 1}^{鈭�} (a_{k} + b_{k})$ diverges.

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