= Integer coefficients with a prescribed radius of convergence
{title2=$a_n=\lceil R^{-n}\rceil\quad(0<R\leq1)$}
Every $R\in[0,1]$ occurs as the <radius of convergence> of a <power series> all of whose <coefficients> are positive <integers>. For $0<R\leq1$, use the <ceiling function> to choose $a_n=\lceil R^{-n}\rceil$. Then
$$
R^{-n}\leq a_n< R^{-n}+1\leq2R^{-n}.
$$
For $|z|<R$, the <comparison test for series> with a <geometric series> proves <absolute convergence>. For $|z|\geq R$, the bound $|a_nz^n|\geq(|z|/R)^n\geq1$ violates the fact that <terms of a convergent series tend to zero>. Hence the <radius of convergence> is exactly $R$.
For $R=0$, use $a_n=n!$: at every $z\ne0$ the ratio of successive term <absolute values> is $(n+1)|z|$, eventually exceeding two. The <factorial> <coefficients> therefore give divergence away from zero.
A <power series> with every <coefficient> a positive <integer> cannot have <radius of convergence> greater than one: $a_n\geq1$ makes convergence impossible at $z=1$. If zero <coefficients> are allowed and only finitely many are nonzero, a <polynomial> instead has infinite <radius of convergence>.
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