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Chapter 7: Q. 40 (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.
${鈭�}_{k = 1}^{鈭�} \frac{k^{k}}{k!}$

Short Answer

Expert verified

The given series diverges.

Step by step solution

01

Step 1. Given Information.

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

02

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

By using the root test, let the general term is $a_{k} = \frac{k^{k}}{k!} .$

So,

$蚁 = \lim_{k 鈫� 鈭�} {(a_{k})}^{\frac{1}{k}} 蚁 = \lim_{k 鈫� 鈭�} {(\frac{k^{k}}{k!})}^{\frac{1}{k}} 蚁 = \lim_{k 鈫� 鈭�} (\frac{k}{{(k!)}^{\frac{1}{k}}}) 蚁 = 鈭�$

Since $鈭� > 1,$ by using the root test the given series diverges.

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Most popular questions from this chapter

If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} Where L i s$ a positive finite number, what may we conclude about the two series?

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.

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

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?

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 ${鈭�}_{k = 1}^{鈭�} (a_{k} + b_{k}) = L + M$ .

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.

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