Use the divergence test to determine if the following series diverge or state that the test is inconclusive. 
${\sum }_{n=1}^{\infty }\frac{{10}^n}{n!}$

• A. Divergent
• B. Convergent
• C. Inconclusive

Answer: (C)

Verified step by step guidance

1. Step 1: Recall the divergence test, which states that if the limit of the nth term of a series as n approaches infinity does not equal zero, the series diverges. If the limit equals zero, the test is inconclusive.
2. Step 2: Identify the nth term of the given series. The nth term is \( \frac{10^n}{n!} \).
3. Step 3: Compute the limit of the nth term as \( n \to \infty \). This involves evaluating \( \lim_{n \to \infty} \frac{10^n}{n!} \).
4. Step 4: Observe the behavior of \( n! \) (factorial) compared to \( 10^n \) as \( n \to \infty \). Factorials grow much faster than exponential functions, so \( \frac{10^n}{n!} \to 0 \).
5. Step 5: Conclude that since the limit of the nth term equals zero, the divergence test is inconclusive for this series. Further tests are required to determine convergence or divergence.