An elementary proof of a generalization of Bernoulli's formula (Q974763)

Let \((B_k)_{k\geq 0}\) be the Bernoulli numbers. Let \(a>-1\) be a real number, \(a=m+\gamma\), \(m\in{\mathbb Z}\), \(-1<\gamma\leq 0\), and let \[ F_a(N)=\sum_{k=0}^m(-1)^k\binom{a+1}{k}B_kN^{m-k+1}. \] The authors prove that there exists a real number \(C_a\) so that \[ \lim_{N\to\infty}\left[(a+1)\sum_{n=1}^Nn^a-N^\gamma F_a(N)\right]=C_a. \] If \(a\geq 0\) is an integer then the sequence on the left-hand side is identically zero.