Using the ratio test to show whether the below series converges or diverges.?

Question

1) show that the power series limit as n appoaches infinity[(x+2)^n!/n^2] converges absolutely for each x such that -3-1.

Answer 1

Taking absolute values a_n= Ix+2I^n! /n^2

if we take a_n+1 /a_n= Ix+2I ^[(n+1)!-n! ] n^2/n+1)^2=
Ix+2I^n(n!) *n^2/(n+1)^2

If Ix+2I>1 lim a_n+1/a_n= +infinity so
a_n+1>a_n and The power serie diverges because lim a_n is not zero
Ix+2I>1 if x>-2 implies x+2>1 so __x>-1 or if x<-2

-2-x>1 x<-3
If Ix+2I<1 lim a_n+1/a_n= 0 and S converges absolutely

-3