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My question is this:

If $\limsup\limits_{n\rightarrow\infty}|a_n|=0$, does it follow that $\limsup\limits_{n\rightarrow\infty}|a_n|^{1/n}=0$?

If not can someone give an example? Thank you, in advance.

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    $\begingroup$ Is it $n\to 0$ or $n\to \infty$? $\endgroup$
    – edm
    Jun 5 '17 at 5:05
  • $\begingroup$ $n \in \mathbb N$ or $\mathbb R^+$, is $a_n$ a $\mathbb R \rightarrow \mathbb R$ function? $\endgroup$
    – Mudream
    Jun 5 '17 at 5:06
  • $\begingroup$ Oh I'm sorry. Edit now. $\endgroup$
    – Xaviere
    Jun 5 '17 at 5:08
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Consider the sequence $a_n=(\frac{1}{2})^n$.

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Let $a_n = \frac{1}{n}$. Then, $a_n \to 0$ (so in particular $\limsup a_n = 0$). But $n^{1/n} \to 1$ as $n \to \infty$, so $a_n^{1/n} \to 1$ as $n \to \infty$.

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Let $a_n=2^{-n}$. Then $$\limsup_{n\to\infty}|a_n|=\lim_{n\to\infty}a_n=0$$ but $a_n^{\frac{1}{n}}=\frac{1}{2}$ for all $n$, so $$\limsup_{n\to\infty}|a_n|^{\frac{1}{n}}=\lim_{n\to\infty}a_n^{\frac{1}{n}}=\frac{1}{2}$$

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