On a result of Bernstein (Q1097427)

According to Bernstein [\textit{A. F. Timan}, Theory of approximation of functions of a real variable (1963; Zbl 0117.290), p. 90] the smallest uniform error obtained in approximating \((1-x)^{-1}\) on [- \[ , \] ] by polynomials \(\sum^{n}_{k=0}c_ kx^ k\), \(n\geq 0\), \(c_ k\) integers, \(c_ n=1\), is \(2^{-n}\). A related result is obtained.