An asymptotic expansion is obtained which provides upper and lower bounds for the error of the best \(L_ 2\) polynomial approximation of degree n for \(x^{n+1}\) on [-1,1]. Because the expansion proceeds in only even powers of the reciprocal of the large variable, and the error made by truncating the expansion is numerically less than, and has the same sign as the first neglected term, very good bounds can be obtained. Via a result of Phillips, these results can be extended from \(x^{n+1}\) to any \(f\in C^{n+1}[-1,1]\), provided upper and lower bounds for the modulus of \(f^{(n+1)}\) are available.