Finding the Radius of Convergence for a Series with Exponential Growth

[Mathmos6]

Hi there - I'm trying to work out the radius of convergence of the series $\sum_{n \geq 1} n^{\sqrt{n}}z^n$ and I'm not really sure where to get going - I've tried using the ratio test and got (not very far) with $lim_{n \to \infty} | \frac{n^{\sqrt{n}}}{(n+1)^{\sqrt{n+1}}}|$, and with the root test, $\left( {lim sup_{n \to \infty} n^{\frac{1}{\sqrt{n}}}}\right) ^{-1}$, neither of which seem to help me =/

I have a strong feeling the latter converges to 1 but even if I'm right I'm not totally sure how to prove it, and I may well be wrong. What should my next move be?

Thanks a lot!

Mathmos6

 

[Dick]

Don't give up so fast on the root test. What IS the limit n->infinity n^(1/sqrt(n))?

 

[Mathmos6]

I'm guessing 1, but I'm not sure how to prove it?

 

[Dick]

I'm guessing 1, but I'm not sure how to prove it?

Take the log to turn it into a quotient. Try to find the limit of the log. Now you can use things like l'Hopital's theorem.

 

[Mathmos6]

That's brilliant, thanks! :)