Awesome problem! I haven't messed with series since calculus, (:. To make the problem a little more clear, I would state U(n, x) as a series and replace domain of convergence with radius of convergence. The main reason is because solving series and solving sequences are much diffrent. As for sequences you are just taking the limit. When you get into series, you are checking whether the series is convergent(curve eventually flattens out to a horizontal line at some number) or divergent(curve shoots up or down to infinity). I liked the idea of your problem and it's kind of unbounded which is awesome. For instance, you could reference a taylor series representation of a function and use that to solve for x. I'm getting ahead of myself but kudos on making the problem.