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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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    If you want an entire characterization of such functions, I'd be surprised if something exists independent of $g$. Off the top of my head it seems like $f(k)=c\cdot k$ for any nonzero constant $c$ works. But in general if $g(k)=0$ for instance, I don't see a nice classification coming about.2012-12-25
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    Sorry I meant $f(k)=c$, not $f(k)=c\cdot k$.2012-12-25
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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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