Convergence of Power Series $\sum_{n=0}^{\infty}\frac{1+\alpha^{n}}{1+\beta^{n}}z^{n}$ with $\alpha ,\beta \ge0$

Question

For $\alpha, \beta \geq 0 \in \mathbb{R}$, find the radius of convergence for the series: $$\sum_{n=0}^{\infty}\frac{1+\alpha^{n}}{1+\beta^{n}}z^{n}$$

Ok, so if $\alpha$ and $\beta$ are $\leq 1$ then it should be pretty easy to find using the root test: $|z|<\rho:=\displaystyle\frac{1}{\displaystyle\limsup_{k\to\infty}\sqrt[k]{|a_{k}|}}$, but how do you do it when $\alpha$ and $\beta$ are $>1\enspace $ ?

Or is my approach to the problem completely wrong? Any answers would be much appreciated!

Answer 1

$$ |z|<\rho:=\displaystyle\frac{1}{\displaystyle\limsup_{k\to\infty}\sqrt[k]{|a_{k}|}} $$ for $\alpha$, $\beta>1$ is equal to $\dfrac{\beta}{\alpha}$, so it is even easier case. Now the case, when they are on different sides of 1, should be also easy.