[math] \newcommand{\mathds}{\mathbb}[/math]

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Develop some convergence theory for [math]x_n=a^n[/math] with [math]a \gt 0[/math], notably by proving that [math]a^n/n^k\to\infty[/math] for any [math]a \gt 1[/math], and any [math]k\in\mathbb N[/math].