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Chapter 7: Q. 74 (page 593)

Determine whether the sequences in Exercises 63鈥�74 are monotonic or not. Also determine whether the given sequence is bounded or unbounded.
$\{5 (1 - {(\frac{1}{10})}^{k})\}$

Short Answer

Expert verified

The given sequence is a monotonic sequence and a bounded sequence.

Step by step solution

01

Step 1. Given Information.

The given sequence is $\{5 (1 - {(\frac{1}{10})}^{k})\} .$

02

Step 2. Determine whether the sequences are monotonic or not.

To determine whether the sequences are monotonic or not, we will use the difference test.

Let the general term is $a_{k} = 5 (1 - {(\frac{1}{10})}^{k}) .$

So, the $a_{k + 1}$ term is

$a_{k + 1} = 5 (1 - {(\frac{1}{10})}^{k + 1}) .$

According to the difference test,

$a_{k + 1} - a_{k} = 5 (1 - {(\frac{1}{10})}^{k + 1}) - 5 (1 - {(\frac{1}{10})}^{k}) = 5 [1 - {(\frac{1}{10})}^{k + 1} - 1 + {(\frac{1}{10})}^{k}] = 5 [{(\frac{1}{10})}^{k} - {(\frac{1}{10})}^{k + 1}] = 5 {(\frac{1}{10})}^{k} [1 - \frac{1}{10}] = 5 {(\frac{1}{10})}^{k} (\frac{9}{10})$

Since $a_{k + 1} - a_{k} > 0 for k > 0 .$

Thus, the sequence is strictly increasing and it is monotonic.

03

Step 3. Determine whether the given sequence is bounded or unbounded.

From the given sequence, we can depict that it is bounded below by zero and bounded above by $5 .$

Thus, the sequence is bounded.

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Most popular questions from this chapter

The contrapositive: What is the contrapositive of the implication 鈥淚f A, then B.鈥�?

Find the contrapositives of the following implications:

If a quadrilateral is a square, then it is a rectangle.

Find an example of a continuous function f : $[1, 鈭�) 鈫� R$ such that ${鈭�}_{1}^{鈭�} f (x) d x$ diverges and localid="1649077247585" ${鈭�}_{k = 1}^{鈭�} f (k)$ converges.

Use the divergence test to analyze the given series. Each answer should either be the series diverges or the divergence test fails, along with the reason for your answer.

${鈭�}_{k = 1}^{鈭�} 鈥� {(\frac{k}{1 + k})}^{k^{2}}$

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

Let ${鈭�}_{k = 0}^{鈭�} c r^{k}$ and ${鈭�}_{k = 0}^{鈭�} b v^{k}$ be two convergent geometric series. Prove that ${鈭�}_{k = 0}^{鈭�} (c r^{k} . b v^{k})$ converges. If neither c nor b is 0, could the series be $({鈭�}_{k = 0}^{鈭�} c r^{k}) ({鈭�}_{k = 0}^{鈭�} b v^{k})$ ?

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