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‎‎Let $f:[1‎, +‎\infty)\rightarrow\mathbb{R}$ be a function such that ‎$‎\lim_{n\to\infty} (f(n) - f(n+1)) = 0‎$‎‎. ‎Suppose that $\lambda:[1‎, +‎\infty)\rightarrow\mathbb{R}$‎ ‎is an exp-convex function satisfying the functional equation $\lambda(x) = f(x)‎ + \lambda‎(x-1)$ for $x\geq 1$ and initial condition $\lambda(0) = 0$‎. ‎Then ‎$‎\lim_{n\to\infty} (xf(n) + \sum_{k=1}^n f(k) - f(k+x))‎$‎ is‎ ‎convergent for $x\geq 1$ ‎, ‎and $‎\lim_{n\to\infty} (xf(n) + \sum_{k=1}^n f(k) - f(k+x)) = \lambda(x)$‎. Thanks.

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closed as unclear what you're asking by Crostul, Yanior Weg, mrtaurho, zz20s, GNUSupporter 8964民主女神 地下教會 May 4 at 14:42

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