Question

Determine whether the sequences in Exercises 63–74 are monotonic or not. Also determine whether the given sequence is bounded or unbounded.

$\left\{\frac{\left(2k\right)!}{{\left(k!\right)}^2}\right\}$

Short answer

The given sequence is monotonic and unbounded.

Step-by-step solution

Step 1. Given Information. 

The given sequence is$\left\{\frac{\left(2k\right)!}{{\left(k!\right)}^2}\right\}.$

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

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

Let the general term of the sequence is $a_k=\frac{\left(2k\right)!}{{\left(k!\right)}^2}.$

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

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

According to the ratio test,

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

Now, $\frac{a_{k+1}}{a_k}>1\mathrm{for}k>0.$

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

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. Thus, there is no bounded above. Hence the sequence is unbounded.