I'm trying to find the radius of convergence of the following complex series: $$\sum^\infty_{n=10}n(z-1)^{2n}$$
I've started by rewriting the above series as follows : $$\sum^\infty_{n=20}\frac{n}{2}(z-1)^{n}$$
This makes it a bit clearer that the radius of convergence $R$ can be found using :
$$\frac{1}{R} = \lim_{n \to \infty }\sup{\sqrt[n]{a_n}} = \lim_{n \to \infty }{\sup \left ({\frac{n}{2}}\right)^\frac{1}{n}} $$
So, the problem is reduced to evaluating the above limit superior. How do I proceed to evaluate this limit?
Some thoughts
- I'm under the impression that squeeze theorem could be useful here.
- Aiming for the above, could I use some inequality between $\lim\sup{\sqrt[n]{a_n}}$ and $\lim{\sqrt[n]{a_n}}?$