I need to find the limit of {(1 + $\displaystyle \frac{1}{2n})^n$}, and justify my answer.

I know that $\displaystyle \stackrel{\lim}{n\rightarrow\infty}$ (1 + $\displaystyle \frac{1}{n})^n$ = e.

Can I use $\displaystyle \frac{n}{2}$ for my 'n,' so I get:

$\displaystyle \stackrel{\lim}{n\rightarrow\infty}$ (1 + $\displaystyle \frac{1}{2(\frac{n}{2})})^\frac{n}{2}$

= $\displaystyle \stackrel{\lim}{n\rightarrow\infty}$ (1 + $\displaystyle \frac{1}{n})^\frac{n}{2}$

= $\displaystyle \stackrel{\lim}{n\rightarrow\infty}$ $\displaystyle [(1 + \frac{1}{n})^n]^\frac{1}{2}$

= $\displaystyle \stackrel{\lim}{n\rightarrow\infty}$ $\displaystyle e^\frac{1}{2}$ = $\displaystyle \sqrt{e}$

Or is $\displaystyle \frac{n}{2}$ not a legitimate choice since I am dealing with naturals?

Thanks for your time,

ilikedmath