TITLE: Approximating $x^m$ by exponentials
QUESTION [7 upvotes]: What's a nice explicit example of a sequence of functions $f_n$: $\mathbb{R}\longrightarrow\mathbb{R}$ of the form
$$a_1e^{c_1x}+\ldots + a_ke^{c_kx},\;\;\; a_i, c_i\in\mathbb{R}$$
that converges to $x^m$ uniformly on every $[a, b]$?
One such example is obtainable by taking the power series for $\log$, composing it with $\exp$ and then using some product of series formula $m$ times, but that results in horrible expressions. There may be something simpler since we are allowing the $c_i$ to be arbitrary reals instead of positive integers.

REPLY [6 votes]: Obviously, we have $x=\lim_{n\rightarrow\infty}n(e^{x/n}-1)$, and the convergence is uniform in any finite interval. Raising that to the $m$th power gives
$$
x^m=\lim_{n\rightarrow\infty}n^m(e^{x/n}-1)^m=\lim_{n\rightarrow\infty}\sum^m_{k=0}n^m(-1)^{m-k}\binom{m}{k}e^{kx/n},
$$
and convergence is still uniform in any finite interval.