Let P and Q be two polynomials in K[x, y] with degree at most d, where K is a field. Denoting by R ∈ K[x] the resultant of P and Q with respect to y, we present an algorithm to compute R mod x^k in O˜(kd) arithmetic operations in K, where the O˜ notation indicates that we omit polylogarithmic factors. This is an improvement over state-of-the-art algorithms that require to compute R in O˜(d^3) operations before computing its first k coefficients.