# A generalization of the Euler-Mascheroni constant

Let $f:[1,+\infty)\rightarrow \mathbb{R}$ be a differentiable function. We are dealing with the limit of the sequence $$f(n)-\sum_{k=1}^nf'(k).$$ If $f=\log$, then it is convergent to $-\gamma$ (where $\gamma$ is the Euler-Mascheroni constant). Now,

(a) Are there some criteria for its convergence (by putting some conditions on $f$)?

(b) Does anyone know some references (paper, book, etc.) about it?

• Are you familiar with the Euler-Maclaurin summation formula? – Steven Stadnicki Jun 1 '17 at 6:05
• Yes, I know it. – M.H.Hooshmand Jun 1 '17 at 6:07
• You may be able to build a set of convergence criteria from the error bounds that are part of that formula. – Steven Stadnicki Jun 1 '17 at 6:09