$\forall n \in \mathbb{N}$, let: $$P_n = \left( \begin{matrix} a & 1-a \\ b_n & 1-b_n \end{matrix} \right). $$
Whereby $\{b_n\}_{n \in \mathbb{N}}$ is a monotonically increasing sequence of numbers such that $0<b_n<1 \; \forall n$ and $$\lim_{n \to \infty} b_n = c \quad 0<c<1.$$ Furthermore, $0 < a < 1$ is known.
I am looking to find an expression for the nth matrix product: $$G_n = P_1 P_2 \dots P_n.$$ Does anyone know if this can be done?
Thanks in advance!