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I have problem in determining the convergence of the series $\sum_{n=1}^{\infty}\left(\frac{1}{n}\right)^{1+\frac{1}{n}}$. It seems like it is convergent given that $(1+\frac{1}{n})>1$ for all n, but I still cannot prove it rigorously.

Can anyone help me ??

Andy Brandi
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KHY
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  • Consider $a_n=\frac{1}{n}$ and try to asymptotically compare $a_n$ with the general term of your sum. – Amihai Zivan Jan 01 '13 at 14:42
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    @AmihaiZivan Couldn't one state it's divergent because $\frac{1}{n}\le\left(\frac{1}{n}\right)^{1+\frac{1}{n}}$ for all $n$ and $\sum_{n\ge 1}\frac{1}{n}$ is divergent? – 000 Jan 01 '13 at 14:44
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  • It is divergent (since the harmonic series is divergent).
  • Your inequality is incorrect.
    • – Amihai Zivan Jan 01 '13 at 14:46
  • @AmihaiZivan I informally meant "for all $n$" to mean ${n: n\ge 1}$. My apologies. – 000 Jan 01 '13 at 14:48
  • @Limitless Read http://en.wikipedia.org/wiki/Limit_comparison_test – Amihai Zivan Jan 01 '13 at 14:48
  • @Limitless You still might want to change the inequality since $n^{1+\frac{1}{n}}\geqslant n$. It shows that direct comparison with the harmonic series wont work and you need to calculate the limit of their ratios. – Amihai Zivan Jan 01 '13 at 14:53
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    @AmihaiZivan Argh. I forgot how to do arithmetic. I'm sorry. – 000 Jan 01 '13 at 14:57
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    This question has been answered very recently: http://math.stackexchange.com/q/266547/51594 – Siméon Jan 01 '13 at 15:59
  • http://www.wolframalpha.com/input/?i=sum+1%2F(k%5E(1%2B1%2Fk)),+k%3D1..infinity – jimjim Jun 17 '17 at 10:29