TITLE: How to prove $(1+1/x)^x$ is increasing when $x>0$?
QUESTION [19 upvotes]: Let $F(x)=(1+\frac{1}{x})^x$.
How do we prove $F(x)$ is increasing when $x>0$?

REPLY [4 votes]: It's enough to show $\ln(1+1/x)^x = x\ln(1+1/x)$ is increasing. Letting $h=1/x$, we can write the last expression as
$$\tag 1 \frac{\ln(1+h)-\ln 1}{h}.$$
We want to show $(1)$ increases as $h$ decreases. Now $(1)$ is just the slope of the line through $(1,\ln 1)$ and $(1+h,\ln(1+h)).$ And any concave function has the property that such slopes increase as $h$ decreases. Since $\ln x$ is concave, we're done.