The power series I have is this:
$$ \sum_{n=0}^\infty a_n z^n ~~~\text{such that}~~ \sum_{n=0}^\infty 2^n a_n ~~\text{converges while}~~\sum_{n=0}^\infty (-1)^n2^na_n~~\text{diverges}$$
This is my attempt:
let $u_n = 2^na_n $, then $\limsup \sqrt[n]{|u_n|} = \limsup\sqrt[n]{|2^na_n|} = 2 \limsup\sqrt[n]{|a_n|} = l_1 < 1 $
Also $\limsup\sqrt[n]{|(-1)^n2^na_n|} = 2 \limsup\sqrt[n]{|a_n|} = l_2 > 1$
Then the radius of convergence of $\sum a_n z^n$ is at least $\frac{2}{l_1}$ and at most $\frac{2}{l_2}$. So the radius is between these two values.
I'm not sure if my approach is right because I'm not getting a definite value.
Any help would be much appreciated.
Edit:
Is it true that $l_1 = l_2$? And therefore in this case the radius = $2$.