Q. 36 For each of the sequences in Exe... [FREE SOLUTION]

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Chapter 7: Q. 36 (page 604)

For each of the sequences in Exercises 23鈥�52 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.
$\{{(1 + \frac{1}{k})}^{k}\}$

Short Answer

Expert verified

The given sequence is monotonic, bounded and convergent.

The limit of the sequence is $e$ .

Step by step solution

Step 1. Given Information

We are given the sequence $\{{(1 + \frac{1}{k})}^{k}\}$ and we need to find if the sequence is monotonic, bounded and the limit if it is convergent.

Step 2. Finding monotonic

The general term is $a_{k} = {(1 + \frac{1}{k})}^{k}$ .

The ratio

$\frac{a_{k + 1}}{a_{k}} = \frac{{(1 + \frac{1}{k + 1})}^{k + 1}}{{(1 + \frac{1}{k})}^{k}} = \frac{\frac{{(k + 2)}^{k + 1}}{{(k + 1)}^{k + 1}}}{\frac{{(k + 1)}^{k + 1}}{k^{k + 1}}} = {(\frac{k + 2}{k + 1} . \frac{k}{k + 1})}^{k} . (\frac{k + 2}{k + 1}) > 1 (k > 0) 鈭� a_{k + 1} > a_{k}$

The sequence is strictly increasing so it is monotonic.

Step 3. Finding bounded

The sequence $\{{(1 + \frac{1}{k})}^{k}\}$ is bounded below because $2 < a_{k}$ . As $k > 1, a_{k} 鈮� 3$ . The decreasing sequence has a lower bound and is $2$ and an upper bound $3$ .

Therefore, the given sequence is bounded.

Step 4. Finding the limit

The monotonic decreasing sequence is bounded below and hence convergent.

$\lim_{k 鈫� 鈭�} a_{k} = \lim_{k 鈫� 鈭�} {(1 + \frac{1}{k})}^{k} = e$

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91影视

For each of the sequences in Exercises 23鈥�52 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.
$\{{(1 + \frac{1}{k})}^{k}\}$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding monotonic

Step 3. Finding bounded

Step 4. Finding the limit

One App. One Place for Learning.

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91影视

For each of the sequences in Exercises 23鈥�52 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. 1+1kk

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding monotonic

Step 3. Finding bounded

Step 4. Finding the limit

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Logic and Functions

Theoretical and Mathematical Physics

Discrete Mathematics

Pure Maths

Calculus

Study anywhere. Anytime. Across all devices.

For each of the sequences in Exercises 23鈥�52 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.
$\{{(1 + \frac{1}{k})}^{k}\}$