(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I know that if [itex]a_n := n^{1/n} - 1[/itex], then [itex]\Sigma a_n[/itex] is divergent. I know this (by the integral test) because the integral of [itex]2^{1/n} - 1[/itex] from 1 to infinity is infinite. However, I want to avoid using non-elementary functions (here, the exponential integral) in my proof that this series is divergent.

Can anyone see a way of doing this?

2. Relevant equations/attempted solution

[itex] lim\ sup_{n\to\infty} a_n^{1/n} = 1[/itex], so the root test is inconclusive. Comparison is getting me nowhere. I'm thinking about seeing whether using the Taylor expansion of each term of the sequence shows what I want to show...

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# A_n := n^(1/n) - 1 yields a divergent series

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