Let $Y=\sum_{k\ge 1} 1_{A_k}$ be an infinite sum of the indicators of independent events. We investigate a precise (as opposed to logarithmic) first-order asymptotic behavior of the tail probabilities $\mathbb{P}\{Y\ge n\}$ and the point probabilities $\mathbb{P}\{Y=n\}$ as $n\to\infty$. Our analysis provides a reasonably complete classification of the asymptotic behaviors covering most cases of practical interest. These general results are then applied to specific examples where the success probabilities $r_k:=\mathbb{P}(A_k)$ decay polynomially $r_k\sim ck^{-β}$ or (sub-, super-) exponentially $r_k\sim ce^{-k^β}$, yielding the asymptotic tail and point probabilities in explicit forms. As briefly discussed in the paper, infinite sums of independent indicators arise naturally in numerous settings as diverse as the range of Poissonized samples, the infinite Ginibre point processes and decoupled renewal processes, and records in the $F^α$ scheme. We also explore the connection of our research to the theory of Hayman-admissible functions and the notion of total positivity.