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Chapter 7: Q. 65 (page 604)

Complete the proof of Theorem 7.18 by evaluating the limits of the sequences in Exercises 65 and 66.
Explain why $lim_{k 鈫� 鈭�} 鈥� k^{1 / k}$ is an indeterminate form. Use L鈥橦opital鈥檚 Rule or another valid method to prove that $k^{1 / k} 鈫� 1$

Short Answer

Expert verified

$k^{1 / k} 鈫� 1$

Step by step solution

01

Step 1. Given information

藛k1/k 鈫� 1.

02

Step 2. Proof

$lim_{k 鈫� 鈭�} 鈥� k^{1 / k} = {鈭�}^{0}$

Taking log on both sides $y = k^{1 / k}$ ,

$\begin{aligned} \ln 鈦� y = \frac{\ln 鈦� k}{k} \\ lim_{k 鈫� 鈭�} 鈥� (\ln 鈦� y) = lim_{k 鈫� 鈭�} 鈥� \frac{\ln 鈦� k}{k} \\ = lim_{k 鈫� 鈭�} 鈥� (\frac{\frac{1}{k}}{1}) \\ = 0 \end{aligned}$

Now,

$lim_{k 鈫� 鈭�} 鈥� (y) = lim_{k 鈫� 鈭�} 鈥� (k^{1 / k})$

$\begin{aligned} lim_{k 鈫� 鈭�} 鈥� (y) = lim_{k 鈫� 鈭�} 鈥� (e^{k^{v / 2}}) (Because \ln 鈦� y = \frac{\ln 鈦� k}{k}) \\ = e^{lim_{i 鈫� 鈭�} 鈥� (k^{v / 2})} (Substitution) \\ = e^{0} \\ = 1 (Simplify) \end{aligned}$

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Most popular questions from this chapter

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.

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

True/False:

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: If $a_{k} 鈫� 0$ , then ${鈭�}_{k = 1}^{鈭�} a_{k}$ converges.

(b) True or False: If ${鈭�}_{k = 1}^{鈭�} a_{k}$ converges, then $a_{k} 鈫� 0$ .

(c) True or False: The improper integral ${鈭�}_{1}^{鈭�} f (x) d x$ converges if and only if the series ${鈭�}_{k = 1}^{鈭�} f (k)$ converges.

(d) True or False: The harmonic series converges.

(e) True or False: If $p > 1$ , the series ${鈭�}_{k = 1}^{鈭�} k^{- p}$ converges.

(f) True or False: If $f (x) 鈫� 0$ as $x 鈫� 鈭�$ , then ${鈭�}_{k = 1}^{鈭�} f (k)$ converges.

(g) True or False: If ${鈭�}_{k = 1}^{鈭�} f (k)$ converges, then $f (x) 鈫� 0$ as $x 鈫� 鈭�$ .

(h) True or False: If ${鈭�}_{k = 1}^{鈭�} a_{k} = L$ and ${S_{n}}$ is the sequence of partial sums for the series, then the sequence of remainders ${L - S_{n}}$ converges to $0$ .

If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 鈭� a n d {鈭�}_{k = 1}^{鈭�} a_{k}$ converges, explain why we cannot draw any conclusions about the behavior of ${鈭�}_{k = 1}^{鈭�} b_{k}$ .

The contrapositive: What is the contrapositive of the implication 鈥淚f A, then B.鈥�?

Find the contrapositives of the following implications:

If a quadrilateral is a square, then it is a rectangle.

Determine whether the series ${鈭�}_{k = 0}^{鈭�} \frac{5^{k + 1}}{{(- 6)}^{k}}$ converges or diverges. Give the sum of the convergent series.

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