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> Suppose that $\{(a\_n)^{\frac{1}{n}}\}\subseteq \mathbb{R}$ converges and denote with $\alpha$ its limit ($a\_n>0$ for all $n$). Is it true that $\{(a\_{n+1})^{\frac{1}{n}}\}$ converges to the same limit than the first sequence? Does it even converges necessarely?
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I'm trying to solve a problem and, if this result happens to be true, the problem simplifies a lot. Can you give me hand with this please? Counterexamples or hints of a possible proof would be extremely useful (I've tried a lot, but I have not reached any of those).