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Chapter 7: Q. 56 (page 592)

In Exercises 55鈥� 58 use the ratio test in Theorem 7.6 to analyze the monotonicity of the given sequence.
$\{\sqrt{1 - \frac{1}{k}}\}$

Short Answer

Expert verified

The given sequence is eventually increasing for $k 鈮� 2 .$

Step by step solution

01

Step 1. Given Information.

The given sequence is $\{\sqrt{1 - \frac{1}{k}}\} .$

02

Step 2. Use the ratio test.

To analyze the monotonicity of the given sequence we will use the ratio test.

Let the general term of the sequence is $a_{k} = \sqrt{1 - \frac{1}{k}} .$

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

$a_{k + 1} = \sqrt{1 - \frac{1}{1 + k}} .$

According to the ratio test,

$\frac{a_{k + 1}}{a_{k}} = \frac{\sqrt{1 - \frac{1}{1 + k}}}{\sqrt{1 - \frac{1}{k}}} = \frac{\sqrt{\frac{1 + k - 1}{1 + k}}}{\sqrt{\frac{k - 1}{k}}} = \frac{\sqrt{\frac{k}{1 + k}}}{\sqrt{\frac{k - 1}{k}}} = \sqrt{\frac{k^{2}}{(k + 1) (k - 1)}}$

Now, $\frac{a_{k + 1}}{a_{k}} > 1 for k 鈮� 2 .$

Thus, the given sequence is eventually increasing for $k 鈮� 2 .$

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