We consider systems of linear differential equations with an irregular singular point of Poincaré rank 1 at infinity. It is well known that there is a fundamental system of formal solution vectors and, for each halfplane, a fundamental system of actual solution vectors having the formal ones as asymptotic expansions. These asymptotic expansions (in the sense of Poincaré) describe the behavior of the actual solutions as theindependent variablezgrows indefinitely, but give no precise error bounds for a givenz, if the asymptotic series are truncated afterNterms. In this paper we show that for large values of |z| the best choice ofNis proportional to |z| and that the resulting error terms are exponentially small.