We are asked to determine if the series $\sum_{n=1}^{\infty} \frac{5^n}{n^5}$ converges or diverges.

1. Problem Description

We are asked to determine if the series $\sum_{n=1}^{\infty} \frac{5^n}{n^5}$ converges or diverges.

2. Solution Steps

To determine whether the given series converges or diverges, we can use the ratio test. The ratio test states that for a series $\sum a_n$, we can compute the limit:

$L = \lim_{n \to \infty} |\frac{a_{n+1}}{a_n}|$

If $L < 1$, the series converges absolutely.

If $L > 1$, the series diverges.

If $L = 1$, the test is inconclusive.

In our case, $a_n = \frac{5^n}{n^5}$. So we have:

$a_{n+1} = \frac{5^{n+1}}{(n+1)^5}$

Now we compute the ratio:

$\frac{a_{n+1}}{a_n} = \frac{\frac{5^{n+1}}{(n+1)^5}}{\frac{5^n}{n^5}} = \frac{5^{n+1}}{(n+1)^5} \cdot \frac{n^5}{5^n} = \frac{5^{n+1}}{5^n} \cdot \frac{n^5}{(n+1)^5} = 5 \cdot \frac{n^5}{(n+1)^5} = 5 \cdot (\frac{n}{n+1})^5$

Now we find the limit as $n$ approaches infinity:

$L = \lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = \lim_{n \to \infty} |5 (\frac{n}{n+1})^5| = 5 \cdot \lim_{n \to \infty} (\frac{n}{n+1})^5$

Since $\lim_{n \to \infty} \frac{n}{n+1} = \lim_{n \to \infty} \frac{1}{1+\frac{1}{n}} = \frac{1}{1+0} = 1$, we have:

$L = 5 \cdot (1)^5 = 5 \cdot 1 = 5$

Since $L = 5 > 1$, the series diverges by the ratio test.

3. Final Answer

The series diverges.