Question

In Exercises 35–40 use the root test to analyze whether the given series converges or diverges. If the root test is inconclusive, use a different test to analyze the series.

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

Short answer

The given series is diverges.

Step-by-step solution

Step 1. Given Information.

The given series is$\sum _{k=1}^{\infty }\frac{5^k}{k^5}.$

Step 2. Determine whether the given series converges or diverges.

From the series, we can depict that if $c_k=\frac{5^k}{k^5}$then $c_{k+1}=\frac{5^{k+1}}{{\left(k+1\right)}^5}.$

So,

$$\begin{gathered} \rho =\underset{k\to \infty }{\lim }\left|\frac{c_{k+1}}{c_k}\right| \\ \rho =\underset{k\to \infty }{\lim }\left|\frac{{\displaystyle \frac{5^{k+1}}{{\left(k+1\right)}^5}}}{{\displaystyle \frac{5^k}{k^5}}}\right| \\ \rho =\underset{k\to \infty }{\lim }\left|\frac{5^{k+1}\left(k^5\right)}{5^k{\left(k+1\right)}^5}\right| \\ \rho =\underset{k\to \infty }{\lim }\left|\frac{5k^5}{{\left(k+1\right)}^5}\right| \\ \rho =5\underset{k\to \infty }{\lim }\left|{\left(\frac{k}{k+1}\right)}^5\right| \\ \rho =5\underset{k\to \infty }{\lim }\left|{\left(\frac{1}{1+{\displaystyle \frac{1}{k}}}\right)}^5\right| \\ \rho =5{\left(\frac{1}{1+{\displaystyle \frac{1}{\infty }}}\right)}^5 \\ \rho =5{\left(\frac{1}{1+{\displaystyle 0}}\right)}^5 \\ \rho =5 \end{gathered}$$

Since$5>1$by using the root test the given series is diverging.