How do I show that for every x>0 on has
(1+x)^(1/x)<e
I proved that e is the limit of the equation, but I don't know how to prove it is strictly less than.
e being defined as the limit as x tends to infinity of (1+1/x)^x.
If $\displaystyle f(x) = (1+x)^{1/x}$ is strictly decreasing on $\displaystyle x>0$ and approaches $\displaystyle e$ as $\displaystyle x \rightarrow 0$, then it must be less than $\displaystyle e$ for $\displaystyle x>0$. So I think showing that $\displaystyle f'(x) < 0$ for $\displaystyle x>0$ should suffice.