Find the limit of the following sequence or determine that the sequence diverges. $$ \left\{\left(1+\frac{20}{n}\right)^{n}\right\} $$

Question

UpStudy AI · Accepted Answer

We are given the sequence

$$
\left\{\left(1+\frac{20}{n}\right)^{n}\right\}
$$

and we want to compute

$$
\lim_{n\to\infty}\left(1+\frac{20}{n}\right)^n.
$$

**Step 1. Recognize the form of the limit.**

The limit is of the form

$$
\lim_{n\to\infty}\left(1+\frac{a}{n}\right)^n,
$$

where in this case $$ a = 20 $$.

**Step 2. Apply the standard limit result.**

It is known that

$$
\lim_{n\to\infty}\left(1+\frac{a}{n}\right)^n = e^a.
$$

Substitute $$ a = 20 $$:

$$
\lim_{n\to\infty}\left(1+\frac{20}{n}\right)^n = e^{20}.
$$

**Step 3. Conclude the result.**

Hence, the limit of the sequence is

$$
e^{20}.
$$
The limit of the sequence $$ \left(1+\frac{20}{n}\right)^n $$ as $$ n $$ approaches infinity is $$ e^{20} $$.

Find the limit of the following sequence or determine that the sequence diverges. \( \left\{\left(1+\frac{20}{n}\right)^{n}\right\} \)