# Convergence or divergence of the series $\sum_{n=1}^{\infty}\left(\frac{1}{n}\right)^{1+\frac{1}{n}}$

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 ??

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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 1 '13 at 14:42
@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 1 '13 at 14:44
1. It is divergent (since the harmonic series is divergent). 2. Your inequality is incorrect. – Amihai Zivan Jan 1 '13 at 14:46
@AmihaiZivan Argh. I forgot how to do arithmetic. I'm sorry. – 000 Jan 1 '13 at 14:57
This question has been answered very recently: math.stackexchange.com/q/266547/51594 – Siméon Jan 1 '13 at 15:59

## 1 Answer

Note that

$$\lim_{n\to\infty} n^{1/n} = 1.$$

Thus we have

$$\lim_{n\to\infty} \frac{\left(\dfrac{1}{n^{1+1/n}}\right)}{\left(\dfrac{1}{n}\right)}= 1.$$

Now you can apply the limit comparison test to conclude that the series diverge to $+\infty$.

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