Q. 62 - Calculus

Question

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

$\left\{\frac{k!}{\left(k+1\right)!}\right\}$

Step-by-Step Solution

Answer

The given sequence is strictly decreasing.  

Step 1. Given Information.

The given sequence is $\left\{\frac{k!}{\left(k+1\right)!}\right\}.$

Step 2. Use the derivative test.

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

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

According to the derivative test,

$$\begin{gathered} f\left(k\right)=\frac{k!}{(k+1)k!} \\ f\left(k\right)=\frac{1}{k+1} \\ f\text{'}\left(k\right)=\frac{\left(k+1\right)\left(0\right)-1\left(1\right)}{{\left(k+1\right)}^2} \\ f\text{'}\left(k\right)=-\frac{1}{{\left(k+1\right)}^2} \end{gathered}$$

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

Therefore, the given sequence is strictly decreasing.