<s>[INST] Proof that $(1+1/x)^x$ is monotonic increasing How does one prove that $(1+\frac{1}{x})^x$ is monotonic increasing for any $x \in [1,\infty)$?
Thanks a million!
 [/INST] $$f(x) = \left ( 1+ \frac{1}{x} \right )^x$$
$$\log{f(x)} = x \log{\left( 1+ \frac{1}{x} \right )}$$
$$\frac{d}{dx}\log{f(x)} = \log{\left( 1+ \frac{1}{x} \right )} + \frac{x}{1+\frac{1}{x}}\left (\frac{-1}{x^2} \right ) = \log{\left( 1+ \frac{1}{x} \right )} - \frac{1/x}{1+ \frac{1}{x}}$$
or
$$\frac{d}{dx}\log{f(x)} = \log{\left( 1+ \frac{1}{x} \right )} + \frac{1}{1+ \frac{1}{x}} - 1 $$
Because $y \log{y} \ge y-1 \; \forall y \ge 1$, we may say that, $\forall x \ge 1$:
$$\frac{d}{dx}\log{f(x)} =\log{\left( 1+ \frac{1}{x} \right )} + \frac{1}{1+ \frac{1}{x}} - 1  \ge 0$$
and $f(x)$ is monotone increasing for all $x \ge 1$, as $\log$ is monotone increasing.
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