asymptotic expansion | semiconvergent series

An asymptotic series is a series expansion of a function in a variable x which may converge or diverge, but whose partial sums can be made an arbitrarily good approximation to a given function for large enough x. To form an asymptotic series R(x) of f(x)~R(x), take x^n R_n(x) = x^n[f(x) - S_n(x)], where S_n(x) congruent a_0 + a_1/x + a_2/x^2 + ... + a_n/x^n.