Question

In Exercises 51–54 use the difference test in Theorem 7.6 to analyze the monotonicity of the given sequence.

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

Short answer

The given sequence is strictly decreasing for$k\ge 1.$

Step-by-step solution

Step 1. Given Information.

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

Step 2. Use the difference test. 

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

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

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

$a_{k+1}=\sqrt{k+1}-\sqrt{\left(k+1\right)+1}.$

According to the test,

role="math" localid="1649229948859" $$\begin{gathered} a_{k+1}-a_k \\ \left(\sqrt{k+1}-\sqrt{k+2}\right)-\left(\sqrt{k}-\sqrt{k+1}\right) \\ \sqrt{k+1}-\sqrt{k+2}-\sqrt{k}+\sqrt{k+1} \\ 2\sqrt{k+1}-\sqrt{k+2}-\sqrt{k} \end{gathered}$$

Now, $a_{k+1}-a_k<0\mathrm{for}\mathrm{all}k\ge 1.$

Thus, the sequence is strictly decreasing for$k\ge 1.$