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