R.O.C of $sum_{0}^{infty} a_n x^n$

Consider the power series $sum_{n =0}^{infty} a_n x^n$. Where $a_0= 0$ and $a_n = frac{sin n!}{n!}$ for $n geq 1$. Let R be the radius of convergence of the power series. Then

  1. $R geq 1$

  2. $R geq 2π$

  3. $R leq 4π$

  4. $R geq π$

My Attempt:

$1over R$ = $lim_{n to infty}(a_n)^{1over n}$ = $lim_{n to infty}frac{a_{n+1}}{a_n}$ = $lim_{n to infty}left(frac{sin (n+1)!}{(n+1)!} over frac{sin n!}{n!}right)$ = $1over1$ = $1$. So options 1,2,4 are true.