# How to solve <munder> <mo form="prefix">lim <mrow class="MJX-TeXAtom-ORD"> n

How to solve $\underset{n\to \mathrm{\infty }}{lim}{\left(\frac{{a}^{\frac{1}{n}}+{b}^{\frac{1}{n}}}{2}\right)}^{n}$
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plodno8n
Assuming $a,b>0$
We have
${\left(\frac{{a}^{\frac{1}{n}}+{b}^{\frac{1}{n}}}{2}\right)}^{n}=a\phantom{\rule{thinmathspace}{0ex}}{\left(\frac{1+{r}^{\frac{1}{n}}}{2}\right)}^{n},$
where $r=b/a$. We can write
$\begin{array}{rl}a\phantom{\rule{thinmathspace}{0ex}}{\left(\frac{1+{r}^{\frac{1}{n}}}{2}\right)}^{n}& =a\phantom{\rule{thinmathspace}{0ex}}{\left(1-\frac{1-{r}^{\frac{1}{n}}}{2}\right)}^{n}=a\phantom{\rule{thinmathspace}{0ex}}\mathrm{exp}\left[n\phantom{\rule{thinmathspace}{0ex}}\mathrm{log}\left(1-\frac{1-{r}^{\frac{1}{n}}}{2}\right)\right]\\ & =a\phantom{\rule{thinmathspace}{0ex}}\mathrm{exp}\left[n\phantom{\rule{thinmathspace}{0ex}}\mathrm{log}\left(1-\frac{1-{e}^{\frac{1}{n}\mathrm{log}r}}{2}\right)\right]\\ & =a\phantom{\rule{thinmathspace}{0ex}}\mathrm{exp}\left[n\phantom{\rule{thinmathspace}{0ex}}\mathrm{log}\left(1-\frac{-\frac{1}{n}\mathrm{log}r+o\left(\frac{1}{{n}^{2}}\right)}{2}\right)\right]\\ & =a\phantom{\rule{thinmathspace}{0ex}}\mathrm{exp}\left[n\phantom{\rule{thinmathspace}{0ex}}\mathrm{log}\left(1+\frac{1}{2n}\mathrm{log}r+o\left(\frac{1}{{n}^{2}}\right)\right)\right]\\ & =a\phantom{\rule{thinmathspace}{0ex}}\mathrm{exp}\left[\phantom{\rule{thinmathspace}{0ex}}\frac{1}{2}\mathrm{log}r+o\left(\frac{1}{n}\right)\right)\right]\\ & \to a\phantom{\rule{thinmathspace}{0ex}}{e}^{\frac{1}{2}\phantom{\rule{thinmathspace}{0ex}}\mathrm{log}r}=a\phantom{\rule{thinmathspace}{0ex}}{r}^{1/2}=\sqrt{ab}.\end{array}$