# Math Help - Sequence problem

1. ## Sequence problem

Find the limit of the sequence.

$a_n = (1 + \frac{7}{n})^{4n}$

$\lim_{n \rightarrow \infty}$ $a_n = ?$

2. $
a_n = (1 + \frac{7}{n})^{4n}
$

$a_n=((1+\frac{7}{n})^n)^4 \rightarrow \ (e^7)^4=e^{28}$

3. Originally Posted by Showcase_22
$
a_n = (1 + \frac{7}{n})^{4n}
$

$a_n=((1+\frac{7}{n})^n)^4 \rightarrow \ (e^7)^4=e^{28}$

Can you show me the steps from

$((1+\frac{7}{n})^n)^4$ to $(e^7)^4$

4. $(1+\frac{x}{n})^n$ converges to $e^x$ as $n \rightarrow \infty$. This is a standard result:

The power series for $e$ is $1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}...... ...=(1+\frac{x}{n})^n$.

In this case, x=7 so we have $(1+\frac{7}{n})^n \rightarrow e^7$