Consider $f(s)=\displaystyle\sum_{i=1}^\infty r_i^s$ where $s\in[0,\infty)$ and $0<r_i<1$. Under what conditions can we claim that there exists some $s$ such that $f(s)=1$.
I know that some $s$ exists in the case of a finite sum (using the facts that $f(s)$ represents a continuous strictly decreasing function over $s\in[0,\infty)$ and $f(0)\geq 1$). However, the infinite case I think is not continuous nor strictly decreasing over $[0,\infty)$. For example, it can be the case that $f(s)=\infty$ over some interval $[0,s_c)$ and $f(s)<\infty$ for all $s\geq s_c$. I think that $f(s)$ should become a continuous strictly decreasing function once it becomes finite. If this is the case, another formulation of the question could be, "Under what conditions can we claim $1\leq f(s_c)\leq \infty$?"