AbstractAn asymptotic expansion of the added mass matrix mA = {mij}i,j=i, 2 of two spheres of radii R1 R2, moving along the line of their centres in an ideal fluid (of zero vorticity) is given for small particle surface distances x (x → 0) to any order in x. The derivation starts from the classical essentially long distance expansion of mA, which is first converted by an inverse Mellin transformation into an integral representation. The expansion is deduced by the shift of the integration line in this representation and by the application of the residue theorem, and its asymptotic character is proved. The pointwise approximation of the matrix mA by a finite number of terms of the asymptotic expansion is investigated in terms of a suitably defined approximation error d(x) for the matrix element mij(x) and its n‐th order approximant m(x) (n = 0, 1,…). The minimum dij(x) of d(x) with respect to the order n is attained for a finite value n = n(x) because of the divergence of the expansion (lim d(x) = ∞ for any x > 0) and defines by it the order n = n(x) of the optimal approximant of mij(x). equation image , which describe the range of x where m(x) approximates mij with an error not larger than ϵ, are introduced, together with the set equation image which gives the distance range x where an approximation of mij(x) by the asymptotic expansion is possible within an error ϵ. By numerical computation the optimal approximation order n(x) and the sets D(ϵ) (n = 1,…, 11) are estimated. It turns out that n(x) is large compared to the highest order (n = 1) approximant m(x) computed previously, whereas the optimal ϵ‐approximation set Dij(ϵ) is typically by one order of magnitude larger than D(ϵ), the ϵ‐approximation set of the approximant m(x).