  Asymptotic approximations ($n \to \infty$) to the truncation errors $r_n = -
\sum_{\nu=0}^{\infty} a_{\nu}$ of infinite series $\sum_{\nu=0}^{\infty}
a_{\nu}$ for special functions are constructed by solving a system of linear
equations. The linear equations follow from an approximative solution of the
inhomogeneous difference equation $\Delta r_n = a_{n+1}$. In the case of the
remainder of the Dirichlet series for the Riemann zeta function, the linear
equations can be solved in closed form, reproducing the corresponding
Euler-Maclaurin formula. In the case of the other series considered -- the
Gaussian hypergeometric series ${}_2 F_1 (a, b; c; z)$ and the divergent
asymptotic inverse power series for the exponential integral $E_1 (z)$ -- the
corresponding linear equations are solved symbolically with the help of Maple.
The practical usefulness of the new formalism is demonstrated by some numerical
examples.
