On a queer binomial sum (Q1265456)

The authors consider the binomial sum \[ S_n(r)=\sum_{m=0}^n{n\choose m}\frac{(-1)^m}{m^r+1} \] with \(r>0\), \(n\in{\mathbb N}\). It turns out that the behaviour of this sum for \(n\to\infty\) depends on the parameter \(r\) and changes dramatically at the values \(r=1\) and \(r=2\). In particular, for \(r\geq 2\) an oscillatory behaviour occurs while for \(r<2\) finally the sequence \(S_n(r)\) becomes monotonous. To be more precise (see Theorem 2), \(S_n(r)\sim\Gamma(r)(\log n)^{-r}\) for \(0<r<1\), \(S_n(r)\sim-\Gamma(r)(\log n)^{-r}\) for \(1<r<2\), and for \(r\geq 2\) it holds that \[ S_n(r)=n^{\cos(\pi/r)}c(r)\cos[a(r)\log n+b(r)]+O(n^{-\delta(r)}) \] with explicitly determined terms \(a(r)\), \(b(r)\), \(c(r)\) and \(\delta(r)\), which are too complicated to be given here. For \(r=1\) the surprisingly simple equation \(S_n(1)=\frac 1{n+1}\) holds. The proofs were established by using Fourier integrals including complex integration depending on the analytical behavior of \(\frac 1{z^r+1}\) for \(z\in{\mathbb C}\).