Show that $(1+x)^{(1+x)}>e^x$ How can I prove that $(1+x)^{(1+x)}>e^x$ for all $x>0$?
The problem arose as I tried to prove the well-known & intuitive econometric principle that the more often you compound your interest, the more interest you ultimately get (in maths, that $\frac{d}{dn}((1+\frac{1}{n})^n)>0$ for $n>0$).
An interesting further problem is to prove that $(a+x)^{(a+x)}>e^x$ is true for all $x>0$ if and only if $a\geq1$.

$$
(1+x)^{1+x} = 1 + x + x^2 + \frac{x^3}{2} + \frac{x^4}{3} + \cdots
$$
is clearly greater than
$$
e^{x} = 1 + x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{24} + \cdots
$$
for $x > 0.$
