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Chapter 7: Q. 12 (page 591)

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

Short Answer

Expert verified

The sequence $\{a_{k}\}$ is eventually decreasing sequence , if it is decreasing for some index $k$ , where $k 鈭� 鈩�$ .

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 decreasing sequence, if it is decreasing for some index $k$ , where $k 鈭� 鈩�$ .

Let $\{a_{k}\}$ be the sequence and $鈩� > 0$ be such that for all $k > 鈩�, a_{k + 1} 鈮� a_{k,}$ then the sequence is eventually decreasing sequence.

03

Step 3. Consider the sequence n-n2n=1∞.

The terms of the sequence can be found by substituting $n = 1, 2, 3, . . .$

Hence, the terms of the sequence are, $1 - 1^{2}, 2 - 2^{2}, 3 - 3^{2}, . . . .$

The terms of the sequence are, $0, - 2, - 6, . . .$

The sequence is eventually decreasing sequence as $a_{2} 鈮� a_{1}, a_{3} 鈮� a_{2}, . . . . ., a_{k + 1} 鈮� a_{k}$ .

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Most popular questions from this chapter

Explain how you could adapt the integral test to analyze a series ${鈭�}_{k = 1}^{鈭�} f (k)$ in which the function $f : [1, 鈭�) 鈫� 鈩�$ is continuous, negative, and increasing.

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.

${鈭�}_{k = 1}^{鈭�} \frac{1}{k}$

Improper Integrals: Determine whether the following improper integrals converge or diverge.

${鈭�}_{1}^{鈭�} \frac{1}{x} d x .$

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

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

Express each of the repeating decimals in Exercises 71鈥�78 as a geometric series and as the quotient of two integers reduced to lowest terms.

$0.237237237 . . .$

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