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?