<s>[INST] How check if the sequence $x_n=(1+\frac{1}{n})^{\frac{1}{n}}$ is monotonically increasing or not? I want to check if the sequence $(1+\frac{1}{n})^{\frac{1}{n}}$ is monotonically increasing. I tried computing $\frac{x_{n+1}}{x_n}$ to check if the ratio is less than 1 or greater than 1, but I am unable to simplify:
$$\frac{x_{n+1}}{x_n} = \frac{n^{\frac{1}{n}}\left(n+2\right)^{\frac{1}{n+1}}}{\left(n+1\right)^{\frac{2n+1}{n\left(n+1\right)}}}$$
I also tried $x_{n+1} - x_n = \left(\frac{n+2}{n+1}\right)^{\frac{1}{n+1}}-\left(\frac{n+1}{n}\right)^{\frac{1}{n}}$
Is there any other way to check the monotonic behaviour of the sequence $x_n$?
Also I would like to know if I can check $y_n=(1-\frac{1}{n})^{\frac{1}{n}}$ using similar arguments?
 [/INST] The sequence decreases if and only if $\frac{\ln\left(1+\frac1n \right)}{n}$ decreases.
let $f(x) = \frac{\ln \left( 1+\frac1x \right)}{x}$ then 
\begin{align}
f'(x) &= \frac{x \cdot \frac{1}{1+\frac1x}\cdot \left( -\frac1{x^2} \right) - \ln \left( 1 +\frac1x \right)}{x^2} \\
&= \frac{-\frac1{x+1}-\ln\left( 1+\frac1x\right)}{x^2}
\end{align}
which is negative for $x>0$.
If we consider $$g(x) = \frac{\ln \left( 1-\frac1x \right)}{x}$$
we have
\begin{align}
g'(x) &= \frac{x \cdot \frac{1}{1-\frac1x}\cdot \left( \frac1{x^2} \right) - \ln \left( 1 -\frac1x \right)}{x^2} \\
&= \frac{\frac1{x-1}-\ln\left( 1-\frac1x\right)}{x^2}
\end{align}
which is positive for $x>1$.

</s>