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Let $\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^ng(k)=A $

Then for what functions $f(x)$ does $\lim_{n\to\infty}\frac{\sum_{k=1}^n f(k)g(k)}{\sum_{k=1}^nf(k)}=A$

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    @Ethan. As Jacob Schlather has said there are many possibilities depending on $g(x)$. Supposing $g(1)\neq g(2)$, if you take $f(x)$ as: f(x) = \left\{ \begin{array}{ll} g(2)-A & \textrm{if $x=1$}\\ A-g(1) & \textrm{if $x=2$}\\ 0 & \textrm{if $x>2$} \end{array} \right. your conditions will be satisfied.2012-12-26

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