  We describe an expansion of Legendre polynomials, analogous to the Taylor
expansion, to approximate arbitrary functions. We show that the polynomial
coefficients in Legendre expansion, therefore the whole series, converge to
zero much more rapidly compared to the Taylor expansion of the same order.
Furthermore, using numerical analysis using sixth-order polynomial expansion,
we demonstrate that the Legendre polynomial approximation yields an error at
least an order of magnitude smaller than the analogous Taylor series
approximation. This strongly suggests that Legendre expansions, instead of
Taylor expansions, should be used when global accuracy is important.
