Consider the following power series $$ S = \sum_{n \geq 0} a_n x^n $$ where $(a_0, a_1, \dotsc)$ is a sequence of complex numbers and $x$ is a complex variable.
If, say, for all $n$ the coefficients $a_n$ satisfy $\lvert a_n \rvert < C$ for some positive real number $C$, then the series $S$ converges for $\lvert x \rvert < 1$. However, if one requires only that $\lvert a_n \rvert < 2^n C$, then for $\lvert x \rvert < \frac12$ the series still converges, but this second constraint is much more relaxed.
I feel that I am missing the necessary (functional?) analysis background to understand the interplay between choosing a fixed $x$ and determining the condition on the sequences $(a_0, a_1, \dotsc)$ for which the series $S$ converges.
More precisely, I would like to determine the (space of?) sequences $(a_0, a_1, \dotsc)$, for which $S$ converges as soon as $0 < \lvert x \rvert < \epsilon \ll 1$.
This follows from the basic theory on power series. Let $A:=(a_n)_{n\geq 0} \subset \mathbb{C}$ be a fixed sequence for which there exists an $\varepsilon>0$ such that the series $$ \sum\limits_{n=0}^\infty a_n z^n \tag{1} $$ converge for $|z|<\varepsilon$. This means that the series $(1)$ has positive radius of convergence. In particular, for the radius $r = \frac{1}{\limsup |a_n|^{1/n}} \geq \varepsilon>0$ (see here for the application of the root test to derive the formula for the radius of convergence).
Thus, the set of all sequences $A$ for which the series $(1)$ converge for some $\varepsilon$ is precisely the set of all sequences $(a_n)$ for which the limit $$ \tag{2} \limsup |a_n|^{1/n} < \infty. $$