What is the radius of convergence of the power series $\sum_{n=1}^\infty \frac{\sin(n!)}{n!} x^n$?

Question

What is the radius of convergence of the power series $$\sum_{n=1}^\infty \frac{\sin(n!)}{n!} x^n$$

I found a question like this and there are four options

1. $R\geq 1$
2. $R\geq 2*\pi$
3. $R\leq 4*\pi$
4. $R\leq \pi$

where $R$ denotes radius of convergence.

I've tried hard but cant get it how to show in equality. Also there is question in my mind that among four options if the series is not converge at the end point can we discard that option. I'm not sure abut this since we know if anywhere "$<$" holds we can say "$\leq$" holds, but the converse isn't not true.

Answer 1

Since $|a_n|\le\frac1{n!}$ it follows that the radius of convergence is $\infty$, just as is the case for $\exp(x)$.