Use the Ratio Test to Investigate the Series $\sum_{n=1}^{\infty} \frac{2^{n}}{n !}$

Multiple Choice

Use the ratio test to investigate the series $\sum_{n=1}^{\infty} \frac{2^{n}}{n !}$ .

A) $\lim _{n \rightarrow \infty} \frac{\frac{2^{n+1}}{(n+1) !}}{\frac{2 n}{n !}}=\lim _{n \rightarrow \infty} \frac{2}{n+1}=0$ , so the series $\sum_{n=1}^{\infty} \frac{2^{n}}{n !}$ diverges.
B) $\lim _{n \rightarrow \infty} \frac{\frac{2^{n+1}}{(n+1) !}}{\frac{2 n}{n !}}=\lim _{n \rightarrow} \frac{2}{n+1}=0<1$ , so the series $\sum_{n=1}^{\infty} \frac{2^{n}}{n !}$ converges.
C) $\lim _{n \rightarrow \infty} \frac{\frac{2^{n+1}}{(n+1) !}}{\frac{2 n}{n !}}=\lim _{n \rightarrow \infty} \frac{2}{n+1}=0$ , so the ratio test fails to tell us anything about the series.
D) $\lim _{n \rightarrow \infty} \frac{\frac{2 n}{n !}}{\frac{2^{n+1}}{(n+1) !}}=\lim _{n \rightarrow \infty} \frac{n+1}{2}=\infty$ , so the series $\sum_{n=1}^{\infty} \frac{2^{n}}{n !}$ converges.