find the limit of the following sequence or determine that the sequence diverges. { ( 1 + 8 / n ) ^ n }…
find the limit of the following sequence or determine that the sequence diverges. { ( 1 + 8 / n ) ^ n } select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the limit of the sequence is. (type an exact answer.) b. the sequence diverges.
Answer
Explanation:
Step1: Recall the limit formula
The well - known limit formula is $\lim_{n\rightarrow\infty}(1 + \frac{a}{n})^n=e^a$. Here $a = 8$.
Step2: Apply the formula
For the sequence $\left{\left(1+\frac{8}{n}\right)^n\right}$, as $n\rightarrow\infty$, we have $\lim_{n\rightarrow\infty}\left(1+\frac{8}{n}\right)^n = e^8$.
Answer:
A. The limit of the sequence is $e^8$