Applying the Ratio Test with Factorials

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.

The series converges absolutely because the factorial in the numerator is dominated by the exponential in the denominator.

The series converges by the Alternating Series Test since the terms alternate in sign.

The Ratio Test is inconclusive for series involving factorials and powers, so further tests are required.