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Chapter 7: Q. 11 (page 591)

Define what it means for a sequence $\{a_{k}\}$ to be eventually strictly increasing.

Short Answer

Expert verified

The sequence $\{a_{k}\}$ is eventually strictly increasing sequence, if it is increasing after some index $k$ ,where $k \in ℕ$ .

Step by step solution

01

Step 1. Given information

The given sequence is, $\{a_{k}\}$ .

02

Step 2. Let ℕ>0.

The sequence $\{a_{k}\}$ is eventually strictly increasing sequence, if it is increasing after some index $k$ , where $k \in ℕ$ .

Let $\{a_{k}\}$ be the sequence and $ℕ > 0$ be such that for all $k > ℕ$ , $a_{k + 1} > a_{k}$ , then the sequence is eventually strictly increasing sequence.

Consider the sequence, $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, . . . ., \frac{n}{n + 1}, . . . .$

The sequence is eventually strictly increasing sequence as $a_{2} > a_{1}, a_{3} > a_{2}, . . . ., a_{k + 1} > a_{k}$ .

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Most popular questions from this chapter

Find the values of x for which the series $\sum_{K = 0}^{\infty} {(\sin x)}^{k}$ converges.

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

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.

$\sum_{k = 1}^{\infty} \frac{k}{3 k^{2} + 100}$

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 $p_{\infty} = h \sum_{k + 1}^{\infty} 0 . 2^{k}$ as k→∞.

(b) Evaluate $p_{\infty}$ .

(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?

Given a series $\sum_{k = 1}^{\infty} a_{k}$ , in general the divergence test is inconclusive when . For a $a_{k} \to 0$ geometric series, however, if the limit of the terms of the series is zero, the series converges. Explain why.

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