Radius of convergence at $2+i$ of $f_i(z)=\frac{1}{\phi_i-2^{1/4}}$ with $2^{1/4}$ being the positive real root

Question

Let $\phi_k(z), k=0,1,2,3$ the branch cuts of $z^{1/4}$. Consider $$f_k(z)=\frac{1}{\phi_k-2^{1/4}}, \quad 2^{1/4}=|2|^{1/4}e^{i0}>0$$ Find the radius of convergence of the series expansion at $2+i$ for each $f_k$

I have several questions:

1. To do this, I think the best way is to see where do I have the singularity for $f_k$, or the the branch cut, because $f_k$ will be analytic elsewhere, so the radius of convergence will be the distance from $2+i$ to the nearest problem. Am I thinking it right?

2. In addition, I guess that I should choose some branch cuts that give me the largest radius of convergence and that are nice, because I could pick a branch cut of the argument in which $f_k(2+i)$ is not defined, or one that takes all complex numbers except for a spiral that goes from $0$ to $\infty$, making the radius of convergence for at least 1 of the root branches as small as I want...

I know then, that $\phi_k=|z|^{1/4}e^{i\frac{arg{z}+2k\pi}{4}}$, $\arg{z} \in (\alpha,\alpha+2\pi)$ for some $\alpha$, what $\alpha$ should I pick? because if I take any $\alpha=2n\pi, n\in\mathbb{Z}$ (this means that I quit the line $\mathbb{R}_{\geq0}$ from my domain). So if I'm understanding good everything about branch cuts... There is no $\phi_k$ that attains the value $2^{1/4}$ (because we would need for $\frac{\arg{z} +2k\pi}{4}=2n\pi$ for some branch cut, but we are removing that value from the domain), but it doesn't matter because every $f_k$ is having radius of convergence $R=1$ since it is the distance from $2+i$ to the positive real line, that is the nearest problem.

3. If instead I quit any other ray that goes form the origin to $\infty$, say $\arg{z} \in (-\pi,\pi)$, then just for one $f_k$ ($f_0$ actually), we will have a singularity at $z=2$, so $R=1$. But for the others $f_k$, we won't have the pole at $z=2$, so $R$ is the distance from $2+i$ to $0$. Am I right?

Please, any help would be of much appreciation as I'm really struggling to understand this properly. I don't know if my reasoning was good or if I have any flaws. The problem doesn't say which branch cuts to consider, so I'm guessing it want me to take one that maximizes the radius $R$ taking into account all $f_k$