Functions with increasing derivatives

[1MileCrash]

Consider a function f(x), such that for all points x0 in the domain, the nth derivative of f evaluated that x0 is less than the n+1th derivative of f evaluated at x0.

A quick example is f(x) = e^(ax) where a > 1, what others are there (not including just changing e to something else)? Is there a name for these?

 

[AlephZero]

It's easy to invent infinite power series that have this property. For example $f(x) = 0 + x + 2x^2 + 3x^3 + \dots$.

Proving (a) it has the required property and (b) it is convergent (for some real values of $x$) are left as exercises for the OP.

Actually, you don't need the infinite series. $f(x) = -1/x$, when $-1 < x < 0$.