A series of the form a₀ + (a₁/x ) + (a₂/x²) + . . . + (a_n/x_n ) + . . . is an asymptotic expansion of the function f (x ) if there exists a number N such that for all n>N the quantity x_n[f (x ) - S_n(x )] approaches zero as x approaches infinity, where S_n(x ) is the sum of the first n terms in the series. Also known as asymptotic series.