# Show that <munder> <mo movablelimits="true" form="prefix">lim <mrow class="MJX-TeXAtom-OR

Show that $\underset{n\to \mathrm{\infty }}{lim}\left[\frac{{n}^{n+1}+\left(n+1{\right)}^{n}}{{n}^{n+1}}{\right]}^{n}={e}^{e}.$ All we have done was elementary manipulations, but we got stuck.
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

lilao8x
Define ${u}_{n}=\left(1+\frac{1}{n}{\right)}^{n}$
${\left(1+\frac{1}{n}{\left(1+\frac{1}{n}\right)}^{n}\right)}^{n}$
$={\left(1+\frac{1}{n}{u}_{n}\right)}^{n}$
$=\mathrm{exp}\left(n\mathrm{log}\left(1+\frac{{u}_{n}}{n}\right)\right)$
$=\mathrm{exp}\left({u}_{n}+\mathcal{O}\left(\frac{1}{n}\right)\right)$The $\mathcal{O}\left({n}^{-1}\right)$ term drops to 0 and ${u}_{n}\to e$ in the limit, so the value sought is $\mathrm{exp}\left(e\right)={e}^{e}$