Notes on some estimations in rational approximation. I (Q1108506)

Several problems in rational approximation are discussed. For instance, an asymptotic expansion is given for \(S_ n=\max | R_ n(x)-e^{- x}|,\) where the maximum is taken over \([0,\infty)\), and \(R_ n(x)=1/\sum^{n}_{k=0}x^ k/k!\) An earlier result \(s_ n<2^{-n}\) is replaced by a sharper estimate. Also, approximations to \((1- x)^{\alpha}\) are considered and Laguerre-Padé approximants. The paper shows an interesting combination of asymptotic and numerical approximations.