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Chapter 7: Q. 71 (page 593)

Determine whether the sequences in Exercises 63鈥�74 are monotonic or not. Also determine whether the given sequence is bounded or unbounded.
$\{\frac{e^{k}}{k!}\}$

Short Answer

Expert verified

The given sequence is eventually decreasing and it is a bounded sequence.

Step by step solution

01

Step 1. Given Information.

The given sequence is $\{\frac{e^{k}}{k!}\} .$

02

Step 2. Determine whether the sequences are monotonic or not.

To determine whether the sequences are monotonic or not, we will use the ratio test.

Let the general term of the sequence is $a_{k} = \frac{e^{k}}{k!} .$

So, the $a_{k + 1}$ term is

$a_{k + 1} = \frac{e^{k + 1}}{(k + 1)!} .$

According to the ratio test,

$\frac{a_{k + 1}}{a_{k}} = \frac{\frac{e^{k + 1}}{(k + 1)!}}{\frac{e^{k}}{k!}} = \frac{(k!) e^{k + 1}}{e^{k} (k + 1)!} = \frac{(k!) e^{k} 路 e}{e^{k} (k + 1) k!} = \frac{e}{k + 1}$

Now, $\frac{a_{k + 1}}{a_{k}} > 1 if 1 < k < e and \frac{a_{k + 1}}{a_{k}} < 1 if k > e .$

Thus, the sequence is eventually decreasing.

03

Step 3. Determine whether the given sequence is bounded or unbounded.

From the sequence, we can depict that it is a bounded sequence.

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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.

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

Leila, in her capacity as a population biologist in Idaho, is trying to figure out how many salmon a local hatchery should release annually in order to revitalize the fishery. She knows that if $p_{k}$ salmon spawn in Redfish Lake in a given year, then only $0.2 p_{k}$ fish will return to the lake from the offspring of that run, because of all the dams on the rivers between the sea and the lake. Thus, if she adds the spawn from h fish, from a hatchery, then the number of fish that return from that run k will be $p_{k + 1} = 0.2 (p_{k} + h) .$ .

(a) Show that the sustained number of fish returning approaches $p_{鈭�} = h {鈭�}_{k + 1}^{鈭�} 0 . 2^{k}$ as k鈫掆垶.

(b) Evaluate $p_{鈭�}$ .

(c) How should Leila choose h, the number of hatchery fish to raise in order to hold the number of fish returning in each run at some constant P?

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.

${鈭�}_{k = 3}^{鈭�} 鈥� \frac{1}{(k 鈭� 2)^{2}}$

Determine whether the series ${鈭�}_{j = 2}^{鈭�} \frac{2^{j}}{3}$ converges or diverges. Give the sum of the convergent series.

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 $0$ .

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