Polynomial-coefficient power series
= Polynomial-coefficient power series
{title2=$sum_{ngeq0}p(n)z^n$}
For a nonzero polynomial $p$, the <power series> $\sum_{n\geq0}p(n)z^n$ has <radius of convergence> $1$. If $p$ has degree $d$ and leading coefficient $c\ne0$, then $p(n)\sim cn^d$ and $|p(n+1)/p(n)|\to1$, so the <ratio test> applies inside the unit disk. At every point of its boundary, the terms fail to tend to zero. The zero polynomial is the exceptional case with <radius of convergence> $\infty$.