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Chapter 7: Q. 54 (page 604)

Determine whether the sequence converges or diverges. If the sequence converges, give the limit.
$\{k^{1 / k!}\}$

Short Answer

Expert verified

Ans: The sequence ${a_{k}} = \{k^{1 / k!}\}$ is divergent.

Step by step solution

01

Step 1. Given information.

given,

$\{k^{1 / k!}\}$

02

Step 2. The objective is to determine whether the sequence is convergent or divergent and to find the limit of the sequence if the sequence is convergent.

In the sequence ${a_{k}} = \{k^{1 / k!}\}$ the general term is $a_{k} = k^{1 / k!}$

The ratio $\frac{a_{k + 1}}{a_{k}}$ gives

$\frac{a_{k + 1}}{a_{k}} = \frac{(k + 1)^{1 / (k + 1)!}}{(k)^{1 / k!}} > 1 (f o r k > 9) T h u s, a_{k + 1} > a_{k}$

The sequence ${a_{k}} = \{k^{1 / k!}\}$ is eventually increasing. The given sequence is monotonic.

03

Step 3. Now,

The sequence ${a_{k}} = \{k^{1 / k!}\}$ is bounded below because

$0 < a_{k} f o r k > 0$

The sequence is an increasing sequence and doesn't have any upper bound.

The given sequence has a lower bound, therefore, the sequence is bounded below.

04

Step 4. The monotonic increasing sequence is bounded above is convergent.

The monotonic decreasing sequence ${a_{k}} = \{k^{1 / k!}\}$ is not bounded above and hence is not convergent. Therefore, the given sequence is divergent.

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Most popular questions from this chapter

Find the values of x for which the series ${鈭�}_{k = 0}^{鈭�} {(\frac{\cos x}{2})}^{k}$ converges.

For each series in Exercises 44鈥�47, do each of the following:

(a) Use the integral test to show that the series converges.

(b) Use the 10th term in the sequence of partial sums to approximate the sum of the series.

(c) Use Theorem 7.31 to find a bound on the tenth remainder, $R_{10}$ .

(d) Use your answers from parts (b) and (c) to find an interval containing the sum of the series.

(e) Find the smallest value of n so that $R_{n} 鈮� 10^{- 6}$

${鈭�}_{k = 1}^{鈭�} \frac{1}{k^{2}}$

Leila finds that there are more factors affecting the number of salmon that return to Redfish Lake than the dams: There are good years and bad years. These happen at random, but they are more or less cyclical, so she models the number of fish $q_{k}$ returning each year as $q_{k + 1} = (0.14 {(鈭� 1)}^{k} + 0.36) (q_{k} + h)$ , where h is the number of fish whose spawn she releases from the hatchery annually.

(a) Show that the sustained number of fish returning in even-numbered years approach approximately $q_{e} = 3 h {鈭�}_{k = 1}^{鈭�} 0 . 11^{k} .$

(Hint: Make a new recurrence by using two steps of the one given.)

(b) Show that the sustained number of fish returning in odd-numbered years approaches approximately $q_{o} = \frac{61}{11} h {鈭�}_{k = 1}^{鈭�} 0 . 11^{k} .$

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

In Exercises 48鈥�51 find all values of p so that the series converges.

${鈭�}_{k = 1}^{鈭�} \frac{k}{{(1 + k^{2})}^{p}}$

Given that $a_{0} = - 3, a_{1} = 5, a_{2} = - 4, a_{3} = 2$ and ${鈭� a_{k}}_{k = 2}^{鈭�} = 7$ , find the value of ${鈭� a_{k}}_{k = 4}^{鈭�}$ .

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