Q. 44 Determine whether the sequence i... [FREE SOLUTION]

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given information.

given,

$\{\frac{k^{2} 鈭� 1}{10000 k + 2}\}$

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

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

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

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

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

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91影视

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

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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91影视

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鈭�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

Discrete Mathematics

Calculus

Pure Maths

Logic and Functions

Decision Maths

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