<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] Considering$$x_n=\left(1+\frac{1}{n}\right)^{\frac{1}{n}}\implies \log(x_n)=\frac{1}{n}\log\left(1+\frac{1}{n}\right)$$
$$\log(x_{n+1})-\log(x_n)=\frac{1}{n+1}\log\left(1+\frac{1}{n+1}\right)-\frac{1}{n}\log\left(1+\frac{1}{n}\right)$$ Now, use Taylor expansion for large values of $n$ to get
$$\log(x_{n+1})-\log(x_n)=-\frac{2}{n^3}+\frac{9}{2 n^4}+O\left(\frac{1}{n^5}\right)$$ Continue with Taylor
$$\frac{x_{n+1}}{x_n}=e^{\log(x_{n+1})-\log(x_n)}=1-\frac{2}{n^3}+\frac{9}{2 n^4}+O\left(\frac{1}{n^5}\right)\implies \frac{x_{n+1}}{x_n} <1 $$
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