Let us suppose that:
$$ f(m) = \sum_{i=0}^{\infty} \overline{p_i}(m) $$
Classical problems in analysis have us knowing $\overline{p_i}(m)$ and questioning ourselves upon the proprieties of $f(m)$: when it is continuous, when it is derivable, etc.
I have the inverse problem: I require that $f(m) \in O(2^m)$ and need to know what functions $\overline{p_i}(m)$ allows for this. I never encountered this kind of problem before (nor heard of it!) so I really do not know where to begin. If someone can give me some pointers, I would be grateful.
Since this problem is originated by me needing some estimates on expected values of aleathory variables, $\overline{p_i}(m)$ represents the probability of failure of the experiment when $m$ objects are involved. This means that I can impose $\overline{p_i}(m) \leq 1 \; \forall i$ and $\overline{p_{i+1}}(m) \leq \overline{p_i}(m) \; \forall m$.