Chapter 7: Q. 44 (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.
$\{\frac{k^{2} - 1}{10000 k + 2}\}$

Short Answer

Expert verified

Ans: $\{\frac{k^{2} - 1}{10000 k + 2}\}$

Step by step solution

Step 1. Given information.

given,

$\{\frac{k^{2} - 1}{10000 k + 2}\}$

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 ${a_{k}} = \{\frac{k^{2} - 1}{10000 k + 2}\}$ the gentral term is $a_{k} = \frac{k^{2} - 1}{10000 k + 2}$ .

Step 3. The general term of the sequence is ak=k2−110000k+2.

The term $a_{k + 1} - a_{k}$ gives

$\begin{aligned} a_{k + 1} - a_{k} = \frac{(k + 1)^{2} - 1}{10000 (k + 1) + 2} - \frac{k^{2} - 1}{10000 k + 2} (Substitution) \\ = \frac{k^{2} + 2 k + 1 - 1}{10000 k + 10000 + 2} - \frac{k^{2} - 1}{10000 k + 2} \\ = \frac{k^{2} + 2 k}{10000 k + 10002} - \frac{k^{2} - 1}{10000 k + 2} (Simplify) \\ = \frac{(k^{2} + 2 k) (10000 k + 2) - (k^{2} - 1) (10000 k + 10002)}{(10000 k + 10002) (10000 k + 2)} \\ = \frac{10000 k^{2} + 10004 k + 10002}{(10000 k + 10002) (10000 k + 2)} \\ > 0 (For k > 0) \\ Thus, a_{k + 1} > a_{k} \end{aligned}$

The sequence ${a_{k}} = \{\frac{k^{2} - 1}{10000 k + 2}\}$ strictly increases. Therefore, the given sequence is monotonic.

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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.
$\{\frac{k^{2} - 1}{10000 k + 2}\}$

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=k2−110000k+2.

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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. k2−110000k+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=k2−110000k+2.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Statistics

Discrete Mathematics

Theoretical and Mathematical Physics

Applied Mathematics

Pure 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.
$\{\frac{k^{2} - 1}{10000 k + 2}\}$