# 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.