Can anyone calculate this limit? (Markov Chains)

If the transitional probabilities are: P11=1/3, P12=2/3, P21=2/3, P22=1/3, P41=1/6 P42=1/6, P43=1/6, P34=1/4, P33=1/4, P31=1/4 What is limP31 as n tends to infinity?

Having drawn the transition diagram i noticed state 1 is recurrent and aperiodic and therefore limP13=fij/mean j as n tends to infinity. Here the mean is the mean recurrence time of j.

However the problem lies in calculating fij=sum from n=1 to infinity of fij(n). Any ideas or suggestions would be very much appreciated. thanks