# It is correct to say that the series $\sum _{n=2}^{\infty}\left( \frac{\log (n)}{n}\right)^n$ converges? [closed]

I found this series but in the convergence proof I don't know if it is correct to say that it converges.

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• What convergence tests have you tried to apply? Sep 22 at 17:59
• This question isn’t at all clear to me. If you’ve seen a proof that it converges, then the problem is…? Sep 22 at 17:59
• Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
– Community Bot
Sep 22 at 18:01
• Show how you concluded the convergence , otherwise it is impossible for us to tell you whether your argument is valid. Sep 22 at 18:03
• But, hint: it can be easier to consider $\log(n)/n$ in terms of $x/e^x$ Sep 22 at 18:07

If $$\displaystyle \lim_{n}\sqrt[n]{a_{n}}=r<1 \,\,\,$$then the series converges.
But $$\displaystyle \lim_{n}\sqrt[n]{(\dfrac{logn}{n}})^{n}$$=$$\displaystyle \lim_{n}\dfrac{logn}{n}= \displaystyle \lim_{n}log\sqrt[n]{n}=log1=0<1$$ . Thus the series converges.