= Power-decay characterization of the spectral radius
{title2=$(x/t)^n\to0\iff t>r(x),\quad t>0$}
In a complex unital <Banach algebra>, the <spectral radius formula> makes the root norms of $(x/t)^n$ tend to $r(x)/t$. If $t>r(x)$, those norms are eventually bounded above by a geometric sequence and the powers vanish. Conversely, vanishing powers eventually have norm less than one; the infimum version of the radius formula then forces $r(x)/t<1$. Consequently $r(x)=\inf\{t>0:(x/t)^n\to0\}$, including the case $r(x)=0$.
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