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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?

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I think your scenario following "of course" is not implied by pointwise convergence: think of f_n(x) as g(x-n), where g is a smooth approximation to the characteristic function of the unit interval. To get the tail small enough, you will need some uniformity in the shrinking of the f_n (or a lot of cancellation). Gerhard "Ask Me About System Design" Paseman, 2013.04.30 –  Gerhard Paseman Apr 30 '13 at 15:13
    
Would the converse not imply there exist values of $x$ for which it is not convergent? Perhaps I'm missing something. It's not essential ($O(1)$ would be enough) but I think it is neater this way. –  Kevin Smith Apr 30 '13 at 15:35
    
Gerhard, thank you. One should note that the scenario I've given in the second paragraph may be taken as the definition of convergence for each $x$. Agreed, the assumptions implicit in my second displayed equation are more restrictive. I do not assume uniformity of convergence because the question is focused on growth rather than behaviour at individual points. –  Kevin Smith May 1 '13 at 11:21
    
I should say, rather than behaviour on compact sets. –  Kevin Smith May 1 '13 at 11:29
    
I like the question. I just think that "of course" should be replaced by "I will assume", since I can come up with examples where your N does not exist. This is the kind of subject Knuth might treat in his Art of Computer Programming Series, and perhaps briefly in his Concrete Mathematics book. I fear that although such methods are practiced on a near daily basis, there is not enough of the subject to warrant a long chapter. If you ask about this within a discipline (e.g. sieve methods in analytic number theory) you might get a hit. Gerhard "And Also You Might Not" Paseman, 2013.05.01 –  Gerhard Paseman May 1 '13 at 16:15
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