Closed-form solution to an infinite series?

Question

I have the following series that I've confirmed on matlab to have some closed-form solution, but I can't find it through trial-and-error, and I definitely don't have the math background to just solve it. Here it is: $$ \sum_{t=0}^\infty \beta^t(1-\delta)^t $$ Where $\beta,\delta\in(0,1)$. Any help would save me oodles of time that I need to spend doing the rest of the problem set (that isn't math problems; this is ancillary to the subject).

Answer 1

This is a geometric series, and $$ \sum_{t=0}^\infty \beta^t(1-\delta)^t =\sum_{t=0}^\infty (\beta(1-\delta))^t =\frac{1}{1-\beta(1-\delta)} $$

Answer 2

This sum converge becouse it is $$=\sum_{i=0}^\infty [\beta(1-\delta)] ^t$$ With argument $\beta(1-\delta) <1$.

If converges to $$\frac{1}{1-\beta(1-\delta)}$$