Find the radius of convergence for the Taylor series of these functions $$ f(z) = \frac{\sqrt{z+2i}}{z-3i}, \quad a=-1 $$ and $$ g(z) = \int_0^z \frac{(t-\sin{t})}{t^2+4}dt, \quad a = 2 $$ As I understand both of these functions are analytic at their respective points $a$, so to find the radius of convergence it suffices to look at the distance to the nearest singularity. For $f(z)$ there are singularities at $z=0$ and $z = 3i$ and hence the radius is $1$.
For $g(z)$ I believe that integrating power series does not change the radius so we can look at the function under the integral sign, which has singularities at $\pm 2i$ and so the radius is $2$. Is this correct or am I going wrong somewhere?
EDIT:
For $f(z)$ I meant that it has a singularity when $z+2i=0$ so at $z=-2i$ which means the radius should be $\sqrt{5}$.