Q. 61 - Calculus

Question

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

$\left\{\frac{\sin k}{k}\right\}$

Step-by-Step Solution

Answer

The test can't analyze the monotonicity of the given sequence. 

Step 1. Given Information.

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

Step 2. Use the derivative test.

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

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

According to the derivative test,

$$\begin{gathered} f\text{'}\left(k\right)=\frac{k\left(\cos k\right)-\sin k}{k^2} \\ f\text{'}\left(k\right)=\frac{k\cos k-\sin k}{k^2} \end{gathered}$$

Now, the sign of $f\text{'}\left(k\right)$ changes an infinite number of times for $k>0.$

Therefore, the test can't analyze the monotonicity of the given sequence.