Chapter 7: Q. 46 (page 604)

Determine whether the sequence is monotonic or eventually monotonic and whether the sequence is bounded above and/or below. If the sequence converges, give the limit.
$\{\tan (\frac{k π}{2})\}$

Short Answer

Expert verified

Ans: The sequence ${a_{k}} = \{\tan (\frac{k π}{2})\}$ is not convergent.

Step by step solution

Step 1. Given information.

given,

$\{\tan (\frac{k π}{2})\}$

Step 2. The objective is to determine whether the sequence is monotonic, bounded above, or bounded below, and to find the limit of the sequence if the sequence is convergent.

In the sequence ${a_{k}} = \{\tan (\frac{k π}{2})\}$ the general term is $a_{k} = \tan (\frac{k π}{2})$ .

Step 3. The general term of the sequence is ak=tan⁡kπ2.

The sequence $a_{k} = \tan (\frac{k π}{2})$ is not monotonic because the sign of $\tan (\frac{k π}{2})$ varies as k increases. Therefore, the given sequence is not monotonic.

Step 4. Now,

The sequence ${a_{k}} = \{\tan (\frac{k π}{2})\}$ is bounded below because

$- \infty < a_{k} < \infty for k > 0$

The given sequence has no lower and upper bounds, therefore, the sequence is not bounded.

Step 5. Thus,

The sequence ${a_{k}} = \{\tan (\frac{k π}{2})\}$ is not monotonic and not bounded. Therefore, the sequence ${a_{k}} = \{\tan (\frac{k π}{2})\}$ is not convergent.

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Determine whether the sequence is monotonic or eventually monotonic and whether the sequence is bounded above and/or below. If the sequence converges, give the limit.
$\{\tan (\frac{k π}{2})\}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. The objective is to determine whether the sequence is monotonic, bounded above, or bounded below, and to find the limit of the sequence if the sequence is convergent.

Step 3. The general term of the sequence is ak=tan⁡kπ2.

Step 4. Now,

Step 5. Thus,

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Determine whether the sequence is monotonic or eventually monotonic and whether the sequence is bounded above and/or below. If the sequence converges, give the limit. tan⁡kπ2

Short Answer

Step by step solution

Step 1. Given information.

Step 2. The objective is to determine whether the sequence is monotonic, bounded above, or bounded below, and to find the limit of the sequence if the sequence is convergent.

Step 3. The general term of the sequence is ak=tan⁡kπ2.

Step 4. Now,

Step 5. Thus,

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Probability and Statistics

Theoretical and Mathematical Physics

Discrete Mathematics

Logic and Functions

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Determine whether the sequence is monotonic or eventually monotonic and whether the sequence is bounded above and/or below. If the sequence converges, give the limit.
$\{\tan (\frac{k π}{2})\}$