# $‎\lim_{n\to\infty} (xf(n) + \sum_{k=1}^n f(k) - f(k+x))‎$‎ is‎ ‎convergent for $x\geq 1$. [closed]

‎‎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.

## 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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