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Let $f$ and $g$ be integrable in $[0,1]$ and $(-\infty, \infty)$ respectively. Let $a_k$ be a divergent series of positive terms and $S_k = a_1 + a_2 + \ldots + a_k$ such that the following asymptotics hold.

$$ \sum_{r=1}^{n} a_rg(S_r) \sim \int_{S_1}^{S_n}g(x)dx $$

and

$$ \sum_{r=1}^{n} a_r f\Big(\frac{S_r}{S_n}\big) \sim S_n \int_{0}^{1} f(x)dx. $$

Given the above information, I am looking for the asymptotics (along with conditions for validity if required) of the following sum.

$$ \sum_{r=1}^{n} a_rg(S_r) f\Big(\frac{S_r}{S_n}\big). $$

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Replace $g$ with $g_n(x)=g(S_n x)$. Then $\mathop{\Sigma}\limits_{r=1}^n a_r g_n(S_r/S_n)\tilde{} S_n \int_0^1 g_n(x)dx$. The lower limit is 0 because the ratio goes to zero in the limit.

Then your final equation is equal to $\mathop{\Sigma}\limits_{r=1}^n a_r g_n(S_r/S_n)f(S_r/S_n)$. This is a Riemann sum (if you replace $a_r$ by $a_r/S_n$), so is asymptotic to $S_n \int_0^1 g_n(x) f(x) dx$.

It's possible to do the opposite and get an integral similar to the first.

This is valid if the functions are Riemann integrable.

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