Use the ratio test to determine whether the series $\sum _{n=1}^{\infty }\frac{\left(n\right)!}{n^n}$ converges or diverges.

• A. The series diverges.
• B. The ratio test is inconclusive.
• C. The series converges.
• D.

Answer: (C)

Verified step by step guidance

1. Step 1: Recall the ratio test. The ratio test states that for a series \( \sum a_n \), if \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L \), then: (a) if \( L < 1 \), the series converges absolutely; (b) if \( L > 1 \), the series diverges; (c) if \( L = 1 \), the test is inconclusive.
2. Step 2: Identify the general term of the series. Here, \( a_n = \frac{n!}{n^n} \). To apply the ratio test, compute \( \frac{a_{n+1}}{a_n} \). Substitute \( a_{n+1} = \frac{(n+1)!}{(n+1)^{n+1}} \) and \( a_n = \frac{n!}{n^n} \) into the ratio.
3. Step 3: Simplify \( \frac{a_{n+1}}{a_n} \). Using the factorial property \( (n+1)! = (n+1) \cdot n! \), the ratio becomes \( \frac{(n+1)!}{(n+1)^{n+1}} \cdot \frac{n^n}{n!} \). Simplify this expression step by step.
4. Step 4: Take the limit as \( n \to \infty \). After simplifying, the ratio \( \frac{a_{n+1}}{a_n} \) will depend on \( n \). Evaluate \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \) to determine the behavior of the series.
5. Step 5: Interpret the result of the limit. If the limit \( L < 1 \), conclude that the series converges absolutely. If \( L > 1 \), conclude that the series diverges. If \( L = 1 \), state that the ratio test is inconclusive.