Use the Ratio Test to determine if the following series converges absolutely or diverges. \[ \sum_{n=1}^{\infty} \frac{4^{n}}{n !} \]

Step 1 ：We are given the series \(\sum_{n=1}^{\infty} \frac{4^{n}}{n!}\). We need to determine whether this series converges or diverges.

Step 2 ：We use the Ratio Test to determine this. The Ratio Test involves taking the limit as n approaches infinity of the absolute value of the ratio of the (n+1)th term to the nth term of the series.

Step 3 ：Let's calculate the ratio of the (n+1)th term to the nth term of the series: \(\frac{n!}{(n+1)!} \cdot \frac{4^{n+1}}{4^{n}}\)

Step 4 ：Simplify the ratio to get \(\frac{4}{n+1}\)

Step 5 ：As n approaches infinity, the ratio simplifies to 0

Step 6 ：Since this value is less than 1, the series converges absolutely according to the Ratio Test.

Step 7 ：Final Answer: The series converges absolutely because \(r=\boxed{0}\)