On rational approximation of \((1-x)^ \sigma\) (Q1924958)

Let \(S_n(x)\) be the \(n\)th partial sum of the power series for \((1-x)^0\) given by: \(S_n(x)= \sum^n_{s=0} (\sigma+j_j-1)x^j\), hold for all \(x\in(0,1)\). Let \(n\geq 1\) and \(0\leq \sigma<1\). The author proves that \[ {{x^\sigma} \over {S_n(x)}} [1-(1-x)^\sigma]< {1\over {S_n(x)}}- (1-x)^\sigma< {{x^\beta} \over {S_n(x)}} [1-(1-x)^\sigma] \] are valid for all \(x\in (0,1)\) if and only if \(\alpha= \infty\) and \(\beta\leq n\) (where \(x^\infty:=0\)).