Use the Ratio Test to determine the convergence or divergence of the series $\sum _ { n = 0 } ^ { \infty } \frac { ( - 6 ) ^ { n } 2 ^ { n } } { n ! }$ .
A)The series converges.
B)The series diverges.

Question

Quizplus · Accepted Answer

The Ratio Test involves taking the limit as $n$ approaches infinity of the absolute value of the ratio of the $(n+1)$th term to the $n$th term of the series. For the given series, the $(n+1)$th term is $\frac{(-6)^{n+1}2^{n+1}}{(n+1)!}$ and the $n$th term is $\frac{(-6)^n2^n}{n!}$. The ratio of these terms simplifies to $\frac{(-6)2}{n+1}$, which approaches 0 as $n$ approaches infinity. Since the limit is less than 1, the Ratio Test confirms that the series converges.

Use the Ratio Test to Determine the Convergence or Divergence ∑n=0∞(−6)n2nn!\sum _ { n = 0 } ^ { \infty } \frac { ( - 6 ) ^ { n } 2 ^ { n } } { n ! }∑n=0∞​n!(−6)n2n​

Use the Ratio Test to Determine the Convergence or Divergence $\sum _ { n = 0 } ^ { \infty } \frac { ( - 6 ) ^ { n } 2 ^ { n } } { n ! }$