Wednesday, December 19, 2018

calculus - Finding limsupnrightarrowinftynfraclog(n)n

I was attempting to show that the power series


n=1nlog(n)zn


has a radius of convergence of 1.


In order to do this I decided to use the α method. This meant evaluating the limit


lim sup


I was able to prove that


\begin{equation*} \lim_{n\rightarrow\infty} \dfrac{\log(n)}{n} = 0 \end{equation*}


but I realized that was necessary but not sufficient to show that the limsup in question is 1.


I then was able to prove that


\begin{equation*} \lim_{n\rightarrow\infty} n^{\frac{1}{n}} = 1 \end{equation*}



However I could not figure out any way of using that fact either.


I realized I could rewrite this limit as


\begin{equation*} \exp\left(\lim_{n\rightarrow\infty} \dfrac{\log(n)^2}{n}\right) \end{equation*}


However since I have not proven L'hôpital's rule, I have no way of evaluating this limit either.


This has left me pretty stuck. I'm not sure how I could tackle this problem from here. Where should I start for this limit? Is there a way to do it without L'hôpital's rule?


I'd really rather not know the whole proof if possible.

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