Zur Konvergenzbeschleunigung von Reihen mit logarithmischer Konvergenz. (Acceleration of the convergence of logarithmically convergent series) (Q2639588)

This paper discusses series \(\sum a_ n\) when \(a_ n=(n^{- \alpha}\ln^{\beta}n)(\gamma_ 0+O(1/n))\) and when \(a_ n=(n^{- 1}\ln^{\beta}n)(\gamma_ 0+\gamma_ 1n^{-1}+O(n^{-2})),\) where Re \(\alpha\) \(>1\), \(\beta\neq 0\) in the first case, Re \(\beta\) \(>1\) in the second case. The author obtains asymptotic expressions for the remainders and then applies a convergence accelerating process. This procedure can be more effective than, for example, the use of the Euler-Maclaurin formula.