does the sequence {an} converge or diverge? find the limit if the sequence is convergent. an=(1 - 1/n)^n…

does the sequence {an} converge or diverge? find the limit if the sequence is convergent. an=(1 - 1/n)^n select the correct choice below and, if necessary, fill in the answer box to complete the choice. a. the sequence converges to lim an = (type an exact answer.) n→∞ b. the sequence diverges.

does the sequence {an} converge or diverge? find the limit if the sequence is convergent. an=(1 - 1/n)^n select the correct choice below and, if necessary, fill in the answer box to complete the choice. a. the sequence converges to lim an = (type an exact answer.) n→∞ b. the sequence diverges.

Answer

Explanation:

Step1: Recall limit formula

We know the well - known limit formula $\lim_{n\rightarrow\infty}(1 + \frac{a}{n})^n=e^{a}$. Here, $a=- 1$ and our sequence is $a_{n}=(1-\frac{1}{n})^n$.

Step2: Calculate the limit

Using the formula $\lim_{n\rightarrow\infty}(1+\frac{a}{n})^n = e^{a}$, when $a = - 1$, we have $\lim_{n\rightarrow\infty}(1-\frac{1}{n})^n=e^{-1}=\frac{1}{e}$. Since the limit exists as $n\rightarrow\infty$, the sequence converges.

Answer:

A. The sequence converges to $\lim_{n\rightarrow\infty}a_{n}=\frac{1}{e}$