Question

Use the Ratio Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{n !}{n 3^{n}} $$

Short answer

The series diverges.

Step-by-step solution

Identify the \( n \)th term of the series

The \( n \)th term of the series is defined as \( a_n = \frac{n !}{n 3^{n}} \).

Find the \( n+1 \)th term of the series

Substitute \( n+1 \) for \( n \) in the function for the \( n \)th term to find the \( n+1 \)th term, \( a_{n+1} = \frac{(n+1) !}{(n+1) 3^{(n+1)}} \).

Calculate the ratio of the \( n+1 \)th term to the \( n \)th term

The ratio \( \frac{a_{n+1}}{a_n} \) evaluates to \( \frac{(n+1) !/3^{n+1}}{n !/3^n} \) which simplifies to \( \frac{n+1}{3} \).

Take the limit of the ratio as \( n \) approaches infinity

Evaluate \( \lim_{n \to \infty} \frac{n+1}{3} \). As \( n \) approaches infinity, the limit of the ratio is infinity.

Use the Ratio Test to determine convergence or divergence

Since the limit is more than 1, by the Ratio Test, the series diverges.