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Chapter 7: Q. 81 (page 605)

Prove the statements about the convergence or divergence of sequences in Exercises 78鈥�83, referring to theorems in the section as necessary. For each of these statements, assume that r is a real number and p is a positive real number.
If $|r| > 1,$ then the sequence $\{r^{k}\}$ diverges and $|r^{k}| 鈫� 鈭�$

Short Answer

Expert verified

Hence proved, $|r| > 1 t h e n r^{k} 鈫� 鈭�$

Step by step solution

01

Step 1. Given information

The given sequence

If $|r| < 1$ then the sequence $\{r^{k}\}$ diverges and ${|r|}^{k} 鈫� 鈭�$

02

Step 2. To prove that the result,use the result of converges of geometric sequence with ratiio less than 1

Since, $|r| > 1$

Therefore,

$\frac{1}{r} < 1,$

The geometric sequence $\{\frac{1}{r^{k}}\}$ with the ration $\frac{1}{r} < 1$ is convergent and converges to 0

Also,if $\lim_{k 鈫� 鈭�} a_{k} = 鈭�,$ then $\frac{1}{a_{k}} 鈫� 0$

The sequence $\{\frac{1}{r^{k}}\}$ is converges to 0.Therefore,

$\lim_{k 鈫� 鈭�} \frac{1}{r^{k}} = 0$

Therefore,

$\lim_{k 鈫� 鈭�} r^{k} = 鈭� f o r |r| > 1$ (Because if $\lim_{k 鈫� 鈭�} a_{k} = 鈭�, t h e n \frac{1}{a_{k}} 鈫� 0)$

Therefore,ifrole="math" localid="1649313485960" $|r| > 1 t h e n r^{k} 鈫� 鈭�$ holds

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Most popular questions from this chapter

Use either the divergence test or the integral test to determine whether the series in Exercises 32鈥�43 converge or diverge. Explain why the series meets the hypotheses of the test you select.

35. ${鈭�}_{k = 1}^{鈭�} \frac{k}{2 + k^{2}}$

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.

${鈭�}_{k = 1}^{鈭�} 鈥� \frac{1}{k^{k}}$

Determine whether the series ${鈭�}_{k = 0}^{鈭�} 蟺 {(\frac{e}{3})}^{k}$ converges or diverges. Give the sum of the convergent series.

$I f \lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 鈭� and {鈭�}_{k = 1}^{鈭�}_____Converges the n {鈭�}_{k = 1}^{鈭�}_____(converges / diverges) .$

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.

${鈭�}_{k = 1}^{鈭�} 鈥� \frac{k!}{k^{k}}$

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