I am trying to compute the limit of the following sum involving a binomial term, where $\mu\in[0,1]$ and $\theta\in[0,1]$:
$ \displaystyle \lim_{t \to \infty} (1-\mu)(1-\theta) \displaystyle \sum_{r=0}^{t-1} {t-1 \choose r} ((1-\mu)(1-\theta))^{t-1-r} \mu^{r} $
The difference compared to standard binomial expressions I see around is that the terms $(1-\mu)(1-\theta)$ and $\mu$ don't add up to 1 necessarily. It only adds up to 1 when $\theta(1-\mu)=0$.
It is clear that this sum is bounded between 0 and 1 and that it does converge to either 0 or 1 depending on the values of $\theta$ and $\mu$. I am not being able to pin this down.
Any help is greatly appreciated!
The sum is $((1-\theta)(1-\mu)+\mu)^{t-1}$ and tends to zero if $\theta>0$ and $\mu<1$. If $\theta=0$ or $\mu=1$, then the limit is $1$. Accordingly the limit of the whole expression is zero if $\theta>0$ and $1-\mu$ if $\theta=0$.