As a background, I'm initially interested in sequences $a_n$ giving rise to functions $\sum_{n=0}^\infty a_nx^n$ for $x\in(0,1)$ and their diverging behavior for $x$ to $1$. E.g. the geometric series $a_n=1$, giving $\sum_{n=1}^\infty x^n=\frac{1}{1-x}$.
Consider $a_n:=n^{-s}$ (including, for negative integers $s$, monomials). For complex $\mid\,x\mid<1$, the polylogarithm $${\mathrm{Li}}_s(x):=\sum_{n=1}^\infty{n^{-s}}x^n$$ converges. Consider now an exponential $a_n:=d^{-n}$ with $d\in(0,1)$. We get
$$\sum_{n=1}^\infty{d^{-n}}x^n=\sum_{n=1}^\infty\left(\dfrac{x}{d}\right)^n$$
Looking at the geometric series, we see that this shrinks the domain for $x$ to $(0,d)$.
Question: What are there for $a_n$ that grow faster than a polynomial in $n$, but still leave the domain for the polylogarithm, at least on the real axis, untouched?