I'm asked to find the radius of convergence for the Taylor series of the following functions (with center provided):
- $f(z) = \frac{e^{z^2}}{(\sin z - 2)^2}$, $z_0 = 1$
- $f(z) = \frac{(z^2 - 1)^2}{(\cos z - 2)^3}$, $z_0 = 1$
- $f(x) = \frac{e^{-x^2}}{(x^2 + 1)^3}$, $x_0 = 1$
- $f(x) = e^{x^4}$, $x_0 = 0$
I think that the radius of convergence is the distance from $z_0$/$x_0$ to the nearest singularities for these functions. For the last function, the radius of convergence is clearly infinity and there's no problem, and even for the third function getting the singularities is straightforward, but for the first two there's problems. I try solving equations such as:
$$(\sin z - 2)^2 = 0 \iff \sin z = 2$$ $$(\cos z - 2)^3 = 0 \iff \cos z = 2$$
I don't know how to find these singularities and thus find the radius of convergence (if I found the satisfying $z$ I could then find their distance to the center and thus get the radius of convergence).