I believe that by the ratio test, your series is absolutely convergent for all x.
using th ratio test looks like this
lim x^(4n+4) 4n! = x^4
n->inf (4n+4)! X^4n (4n+4)(4n+3)(4n+2)(4n+1) so the lim =0 as n approaches infinity which makes this series convergent for all values of x.