  This paper considers the approximation of a monomial $x^n$ over the interval
$[-1,1]$ by a lower-degree polynomial. This polynomial approximation can be
easily computed analytically and is obtained by truncating the analytical
Chebyshev series expansion of $x^n$. The error in the polynomial approximation
in the supremum norm has an exact expression with an interesting probabilistic
interpretation. We use this interpretation along with concentration
inequalities to develop a useful upper bound for the error.
