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

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Convergence n^(1/n)

**Physics Forums | Science Articles, Homework Help, Discussion**