it's (3(n+1))/(n+1)! / 3n/n! not (3n+1)/(n+1)! / 3n/n! you forgot the brackets around n+1, that makes the answer totally different. in fact, it makes the solution unknown by the ratio test, since using the ratio test will give a limit of 1.
the (-1)^n+1 doesn't matter since you took absolute values when doing the test, so you assume it is always positive.
use the alternating series test here