Q. 42 - Calculus

Question

Use either the divergence test or the integral test to determine whether the series in Exercises 32–43 converge or diverge. Explain why the series meets the hypotheses of the test you select.

42. $\sum _{k=1}^{\infty }\frac{2^k}{k^5}$

Step-by-Step Solution

Answer

The series is divergent.

Step 1. Given information

We have been given the series $\sum _{k=1}^{\infty }\frac{2^k}{k^5}$

We have to determine whether the series converge or diverge.

Step 2. Determine whether the series converge or diverge.

The divergence test states that if the sequence $\left\{a_k\right\}$ does not converge to zero, then the series $\sum _{k=1}^{\infty }\frac{2^k}{k^5}$ diverges.

The value of the sequence $\left\{a_k\right\}=\left\{\frac{2^k}{k^5}\right\}$ is:

$$\begin{gathered} \underset{k\to \infty }{\lim }{a}_k \\ =\underset{k\to \infty }{\lim }\left(\frac{2^k}{k^5}\right) \\ =\infty \end{gathered}$$

The series is divergent.