Take for the incomplete gamma function with parameter \(a\) the alternating series [see, e.g., \textit{M. Abramowitz} and \textit{I. A. Stegun}, Handbook of mathematical functions with formulas, graphs, and mathematical tables. (1964; Zbl 0171.385), p. 26] which is, up to an exponential factor and a \((1-a)\)-th power factor, a Laurent series about 0 with only nonpositive- exponent terms. Clearly the \(2n\)-th and the \((2n+1)\)-st partial sums are approximations from two sides. Continuing joint work with \textit{A. Laforgia} [J. Comput. Appl. Math. 23, No. 1, 25-33 (1988; Zbl 0645.33001)], the author improves these estimates by further adjusting the last terms in these sums. The reviewer cannot resist quoting one sentence from p. 388: ``We tAsted the three bounds of \(f(a,x)\)''\dots .