Convergence radius for Taylor series $f(z)=e^{z^3}\sin^3(z)-\frac{1}{2}\cos^2(z)+5\sin(z)$

Question

Find the convergence radius for the Taylor series of $f(z)=e^{z^3}\sin^3(z)-\frac{1}{2}\cos^2(z)+5\sin(z)$.

Since I have no singularities, does the Taylor series converges over all of $\mathbb{C}$, i.e. $R=\infty$?

Answer 1

$f$ is the sum of products of functions which Taylor series' have infinite convergence radius. Namely $e^{z^3}, \sin z,\cos z$. Therefore the convergence radius of $f$ is infinite.

Answer 2

Yes, the function is entire and its radius of convergence is $\infty $.