Q. 59 - Calculus

Question

In Exercises 59–62 use the derivative test in Theorem 7.6 to analyze the monotonicity of the given sequence.

$\left\{\frac{\sqrt{k+1}}{k}\right\}$

Step-by-Step Solution

Answer

The given sequence is strictly decreasing. 

Step 1. Given Information.

The given sequence is $\left\{\frac{\sqrt{k+1}}{k}\right\}.$

Step 2. Use the derivative test.

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

Let the function is $f\left(k\right)=\frac{\sqrt{k+1}}{k}.$

According to the derivative test,

$$\begin{gathered} f\text{'}\left(k\right)=\frac{k{\displaystyle \frac{1}{2\sqrt{k+1}}}-\sqrt{k+1}}{k^2} \\ f\text{'}\left(k\right)=\frac{{\displaystyle \frac{k-2\left(k+1\right)}{2\sqrt{k+1}}}}{k^2} \\ f\text{'}\left(k\right)=\frac{k-2k-2}{\left(k^2\right)2\sqrt{k+1}} \\ f\text{'}\left(k\right)=\frac{-k-2}{\left(k^2\right)2\sqrt{k+1}} \\ f\text{'}\left(k\right)=-\frac{k+2}{\left(k^2\right)2\sqrt{k+1}} \end{gathered}$$

Now, $f\text{'}\left(k\right)<0 \mathrm{for} \mathrm{all} k>0.$

Therefore, the given sequence is strictly decreasing.