In regard to the characteristics of certain "explicit formulae" arising in number theory, I am pondering the connection between the rate of convergence of series and the asymptotic order of the function it represents, and how this connection might be exploited in order to isolate the significant terms of the series. This is one of the most google-unfriendly topics I have tried to research, hence posting this question.
Essentially, I am considering the following notion. Given a pointwise convergent series $f(x)=\sum_{n\geq 1} f_n(x)$, $x>x_0$ say, then of course given $\epsilon$ there exists a function $N(x,\epsilon)$ such that for all $x>x_0$ and $N>N(x,\epsilon)$ one has $$|\sum_{n\geq N}f_n(x)|<\epsilon.$$ So, in principle, one may choose $\epsilon=\epsilon(x)$ and ask:
What proportion of the terms contribute to the asymptotic order of $f$?
Naturally one would like to "use" as few terms as possible. To make this more tangible (though not yet satisfactory and clearly less general), one may write $$\sum_{n\geq y}f_n(x)=O(x^qy^{-p})$$ for some functions $p=p(x)$ and $q=q(y)$, where I am implicitly assuming that $p$ is bounded away from $0$ and $q$ is non-increasing. Choosing, say, $y=x^{\delta}$ gives $$\sum_{n\geq x^{\delta}}f_n(x)=O(x^{q(x^{\delta})-\delta p(x)})$$ and so, given $\epsilon>0$, one may write $$\sum_{n\geq x^{\epsilon/p(x)}}f_n(x)=O(x^{q(1)-\epsilon}).$$
The kind of applications I have in mind are those in which one is given (perhaps for the sake of argument) that $q(1)=A+\epsilon$ but $q(1)\neq A$, for then the above estimate is $O(x^A)$ and it would follow that $$f(x)=\sum_{n\leq x^{\epsilon}}f_n(x)+O(x^A),$$ so it appears that merely an arbitrarily small proportion of the terms are contributing to the asymptotic order of $f$.
Thus, the bottom line of my question is:
Where might I find a rigorous treatment of this line of reasoning, preferably in some more generality than I have given here, or; do you know of any applications where this sort of truncation has been successfully applied to get good results?
