A functional approach to lower bounds for numerical series

In this work, we investigate lower bounds for real-valued series of the form ∑ n ≥ 0 a n , with particular emphasis on the case where the associated linear functional, defined through the sequence of moments ( a n ) n ≥ 0 , is positive. We examine how this positivity condition impacts the derivation of lower bounds for the series, utilizing techniques from functional analysis. By considering both general lower bounds and those dependent on specific properties of the sequence, we provide a comprehensive framework for understanding the convergence and asymptotic behavior of such numerical series. Special attention is given to the role of moment sequences, Hankel determinants, and their connections to the underlying functional structure. The results offer novel insights into the convergence criteria and the possible acceleration of series summation.