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Let a real sequence {$x_n$}$_{n\ge 1}$ be such that $\lim_{n \to \infty} nx_n=0$

Find all real values of t such that $\lim_{n \to \infty} x_n(\log n)^t=0$

My approach is $\log n\to \infty$, so $t > 0$, but I am not sure about it. Also, I am not sure how to write the solution.

Matcha Latte
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Vikas Sharma
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1 Answers1

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Since $\;\lim\limits_{n\to\infty}\cfrac{(\log n)^t}{n}=0\;$ for all $\;t\in\mathbb{R}\;,\;$

$\lim\limits_{n\to\infty}x_n\left(\log n\right)^t=\lim\limits_{n\to\infty}nx_n\cfrac{\left(\log n\right)^t}{n}=$

$=\lim\limits_{n\to\infty}nx_n\cdot\lim\limits_{n\to\infty}\cfrac{(\log n)^t}{n}=0\;,\;$ for all $\;t\in\mathbb{R}\;.$

Angelo
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