A sequence is generated by the following recursion:
$$ A(k) = \frac{1}{1 + {(\beta R)}^{- 1 / \nu} R - {(\beta R)}^{- 1 / \nu} R A(k - 1)} $$
with $A(0) = 0$. Can you help me with a function that generates $A(k)$ non-recursively? Also, what is $\lim_{k \to \infty} A(k)$?
This looks like a linear-fractional dynamical system, of the general form $z^* = \frac{ az+ b}{cz+d}$. There is a standard trick in which the numerator and denominator are treated as two components of a vector $(N,D)$ that are updated using $2 \times 2$ matrix notation as $(N^*,D^*)= (N, D) M$ where $M= \pmatrix{a&c\\b&d }$ contains $a,b,c,d$. The iterates of your recursion are therefore matrix powers of $M$ applied to your initial seed value $(N,D)=(0,1)$. You can compute the iterates of matrix $M$ many ways, such as by first finding its eigenvector basis representation.