Hello, I was wondering how to proof [tex]a_n = n^{1/n} \to 1.[/tex](adsbygoogle = window.adsbygoogle || []).push({});

Doing it straight from the definition got me nowhere. But I was thinking. It is obvious that [tex]\liminf a_n \geq 1[/tex] (since otherwise for big n you could get n^(1/n) < 1 <=> n < 1). And I also have already a proof of [tex]\limsup x_n^{1/n} \leq \limsup \frac{x_{n+1}}{x_n}[/tex] (which is a general result for any row x_n).

But proving [tex]\limsup \frac{a_{n+1}}{a_n} \leq 1[/tex] seemed to be harder than I thought.

So I'm completely stuck. Any ideas?

Thank you,

mr. vodka

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# Convergence n^(1/n)

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