Question

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

$\left\{\frac{k}{k+2}\right\}$

Short answer

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

Step-by-step solution

Step 1. Given Information.

The given sequence is$\left\{\frac{k}{k+2}\right\}.$

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$a_k=\frac{k}{k+2}.$

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

$a_{k+1}=\frac{\left(k+1\right)}{\left(k+1\right)+2}.$

According to the test,

$$\begin{gathered} a_{k+1}-a_k \\ =\frac{k+1}{k+3}-\frac{k}{k+2} \\ =\frac{\left(k+1\right)\left(k+2\right)-k\left(k+3\right)}{\left(k+3\right)\left(k+2\right)} \\ =\frac{k^2+k+2k+2-k^2-3k}{\left(k+3\right)\left(k+2\right)} \\ =\frac{2}{\left(k+3\right)\left(k+2\right)} \end{gathered}$$

Now,role="math" localid="1649230532100" $a_{k+1}-a_k\ge 0\mathrm{for}\mathrm{all}k\ge 1.$

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