power series

Question

 Determine the interval and radius of convergenxe for the power series the summation of x4n (4n)! 

Answer 1

   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.