On asymptotics, Stirling numbers, Gamma function and polylogs

Daniel B. Grünberg

We apply the Euler--Maclaurin formula to find the asymptotic expansion of the sums ${\sum }_{k=1}^n(\log k{)}^p/k^q$, ~$\sum k^q(\log k{)}^p$, ~$\sum (\log k{)}^p/(n-k{)}^q$, ~$\sum 1/k^…