Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 ??

share|cite|improve this question
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: – Siméon Jan 1 '13 at 15:59

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 $+\infty$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.