= Bounded power-series terms force interior absolute convergence
{title2=$|a_nz^n|\le M\ \Longrightarrow\ \sum_n|a_nw^n|<\infty\quad(|w|<|z|)$}
If the terms of a <power series> at a nonzero point $z$ are bounded by $M$, then $|a_nw^n|\le M(|w|/|z|)^n$. A convergent <geometric series> and the <comparison test for series> give <absolute convergence> at every point strictly inside that circle. Consequently terms at every point outside the <radius of convergence> must be unbounded. This conclusion is stronger than merely saying that those terms fail to tend to zero.
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