= Polynomial coefficient growth with unit power-series radius
{title2=$c\le a_n\le C(1+n)^m\Longrightarrow R=1$}
Suppose real coefficients satisfy $c\le a_n\le C(1+n)^m$ for all sufficiently large $n$, where $c,C>0$ and $m\ge0$. Then the <power series> $\sum a_nz^n$ has <radius of convergence> one. For $|z|<1$, the <comparison test for series> and the <ratio test> dominate it by a polynomial times a geometric sequence. For $|z|\ge1$, the terms have <modulus> at least $c$ eventually and violate the <term test for divergence>. Finite changes to the coefficients do not affect the conclusion.
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