Question

In Exercises 29–34 use the ratio test to analyze whether the given series converges or diverges. If the ratio test is inconclusive, use a different test to analyze the series.

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

Short answer

The series diverges.

Step-by-step solution

Step 1. Given information.

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

Step 2. Ratio test.

According to the given series,

$$\begin{gathered} \frac{a_{k+1}}{a_k}=\frac{\frac{5^{k+1}}{(k+1{)}^5}}{\frac{5^k}{(k{)}^5}} \\ =\frac{(k{)}^5{5}^{k+1}}{(k+1{)}^5{5}^k} \\ =\frac{5{\left(k\right)}^5}{(k+1{)}^5} \end{gathered}$$

Step 3. Conclusion.

On taking limits,

$$\begin{gathered} \underset{k\to \infty }{\lim }\frac{a_{k+1}}{a_k}=\underset{k\to \infty }{\lim }\frac{5(k{)}^5}{(k+1{)}^5} \\ =5\left(\underset{k\to \infty }{\lim }\frac{(k{)}^5}{(k+1{)}^5}\right) \\ =5\left(\underset{k\to \infty }{\lim }\frac{(k{)}^5}{(k{)}^5{\left(1+\frac{1}{k}\right)}^5}\right) \\ =5\times 1 \\ =5 \\ \text{Since,}L>1, \end{gathered}$$

Therefore, the series diverges.