Problem 51

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

Verified

Answer

The series diverges.

1Step 1: 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}} $.

2Step 2: 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)}} $.

3Step 3: 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} $.

4Step 4: 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.

5Step 5: Use the Ratio Test to determine convergence or divergence

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