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?

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

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