Chapter 7: Q. 20 (page 624)

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.
$\sum_{k = 1}^{\infty} \frac{1}{k^{k}}$

Short Answer

Expert verified

Ans: The divergence test fails divergent because, $\sum_{k = 1}^{\infty} \frac{1}{k^{k}} = 0$

Step by step solution

Step 1. Given information.

given,

$\sum_{k = 1}^{\infty} \frac{1}{k^{k}}$

Step 2. The objective is to use the divergence test to analyze the given series.

The divergence test states that if the sequence ${a_{k}}$ does not converge to zero, then the series localid="1649063033376" $\sum_{k = 1}^{\infty} a_{k}$ diverges.

The value of the sequence localid="1649063415009" $\{a_{k}\} = \{\frac{1}{k^{k}}\}$ is:

$\begin{aligned} lim_{k \to \infty} a_{k} = lim_{k \to \infty} (\frac{1}{k^{k}}) = 0 \end{aligned}$

The divergence test fails divergent because, $lim_{k \to \infty} (\frac{1}{k^{k}}) = 0$

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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.
$\sum_{k = 1}^{\infty} \frac{1}{k^{k}}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. The objective is to use the divergence test to analyze the given series.

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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. ∑k=1∞ 1kk

Short Answer

Step by step solution

Step 1. Given information.

Step 2. The objective is to use the divergence test to analyze the given series.

One App. One Place for Learning.

Most popular questions from this chapter

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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.
$\sum_{k = 1}^{\infty} \frac{1}{k^{k}}$