APFIVE Applying the Ratio Test with Factorials
Which of the following conclusions is correct regarding the series $$\sum_{n=1}^{\infty} \frac{n!}{2^n}$$ as determined by the Ratio Test?
A
The series converges by the Alternating Series Test since the terms alternate in sign.
B
The series diverges because the Ratio Test yields $$\lim_{n\to\infty} \frac{(n+1)!/2^{n+1}}{n!/2^n} = \lim_{n\to\infty} \frac{n+1}{2} = \infty$$, which is greater than 1.
C
The series converges absolutely because the factorial in the numerator is dominated by the exponential in the denominator.
D
The Ratio Test is inconclusive for series involving factorials and powers, so further tests are required.
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