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Chapter 7: Q. 17 (page 592)

Give an example of a sequence with neither an upper bound nor a lower bound.

Short Answer

Expert verified

Ans: The sequence ${\{(- 1)^{k} k\}}_{k = 0}^{\infty}$ neither has an upper bound nor a lower bound.

Step by step solution

01

Step 1. Given information.

Give an example of a sequence with neither an upper bound nor a lower bound.

02

Step 2. The objective is to give an example of the sequence which has neither an upper bound nor a lower bound.

Consider the sequence ${\{(- 1)^{k} k\}}_{k = 0}^{\infty}$

Find the terms of the sequence by substituting $k = 0, 1, 2, 3, . . . .$

The terms of the sequence are $\{(- 1)^{0} \cdot 0, (- 1)^{1} \cdot 1, (- 1)^{2} \cdot 2, (- 1)^{3} \cdot 3, (- 1)^{4} \cdot 4, \dots \cdot\}$

Hence, the terms of the sequence arerole="math" localid="1649305772298" ${0, - 1, 2, - 3, 4, . . .}$ .

03

Step 3. Thus,

The above sequence is neither an increasing sequence nor a decreasing sequence.

Hence, the sequence neither has an upper bound nor a lower bound.

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Most popular questions from this chapter

Let $\sum_{k = 1}^{\infty} a_{k} b e a c o n v e r g e n t s e r i e s a n d \sum_{k = 1}^{\infty} b_{k} b e a d i v e r g e n t s e r i e s .$ Prove that the series $\sum_{k = 1}^{\infty} (a_{k} + b_{k})$ diverges.

Use either the divergence test or the integral test to determine whether the series in Exercises 32–43 converge or diverge. Explain why the series meets the hypotheses of the test you select.

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

Determine whether the series $\sum_{k = 0}^{\infty} \frac{{(- 3)}^{k + 1}}{4^{k - 2}}$ converges or diverges. Give the sum of the convergent series.

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 ${\sum a_{k}}_{k = 4}^{\infty}$ .

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

$\sum_{k = 2}^{\infty} \frac{1}{k {(\ln (k))}^{p}}$

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