1.
PnP_n is 3×33\times3 with Pr=[aij]P_r = [a_{ij}], aij=3i+j42na_{ij} = \frac{3i+j}{4^{2n}}. Find limnTr(4P1+42P2++4nPn)\lim_{n\to\infty} \text{Tr}(4P_1+4^2P_2+\ldots+4^nP_n)