Evaluation of $\lim_{n\to \infty} \frac{1}{n} \text{ln}(1 - a \frac{ n}{(1 - z)^n} )$ I'm trying to find the limit of the function
$$\frac{1}{n} \text{ln}\left(1 - a \frac{ n}{(1 - z)^n} \right)$$
as $n \to \infty$, where a and z are complex and $|z| < 1$. 
I plugged this expression into Mathematica and after some experimenting found that it is convergent to some complex number, and that the result is actually independent of a, but I haven't been able to find a rigorous reason why. Can anyone see how you would evaluate this?

Very rough answer: of course it is independent of $a$:
$$\frac{1}{n} \log{\left ( 1-a \frac{n}{(1-z)^n}\right)} \sim \frac{\log{(-a)}}{n} + \frac{\log{n}}{n} - \log{(1-z)} \sim -\log{(1-z)}$$
as $n \to \infty$, for $|1-z| < 1$.  Otherwise, if $|1-z|>1$, the expression goes as
$$-\frac{a}{(1-z)^n} \to 0$$
in that limit.  
