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 supn→∞nlog(n)n
I was able to prove that
limn→∞log(n)n=0
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
limn→∞n1n=1
However I could not figure out any way of using that fact either.
I realized I could rewrite this limit as
exp(limn→∞log(n)2n)
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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