Limit of $(1-c^n)^{bn}$ when $n\to\infty$, for $c \in (0,1)$ and b is positive constant (arbitrary large). Limit of $(1-c^n)^{bn}$ when $n\to\infty$. For $c \in (0,1)$ and b is positive constant (arbitrary large).
The inner part grows to 1 exponentially. The outer part, however, decreases the whole value exponentially.
I tried some Matlab simulations, they all converge to 1.
I wonder how to justify the result through mathematical proof.
In addition, is it possible to characterize the convergence speed? Linear (exponential), sublinear, etc.?

I will show that $\lim\limits_{n \to \infty} \log((1-c^n)^{bn})=0$ , the conclusion follows easily.
$$\log((1-c^n)^{bn}) = bn \log(1-c^n) = bn(-c^n) \cdot \frac{\log(1-c^n)}{-c^n} \stackrel{n \to \infty}\to 0\cdot 1 = 0$$
