Infinite series converges to 1 proof

Question

Consider the series

$$S_n = a_1 + a_2(1-a_1) + a_3(1-a_2)(1-a_1)+\cdots+ a_n\prod_{i=1}^{n-1}(1-a_i)$$

$$0

For $a_i=a_j=a\;\; \forall i,j$ it is immediate to show that $S_n$ converges to $1$ as $n\to \infty$.

I also simulated drawing the $a_i$'s as i.i.d. Uniform $U(0,1)$ random variables, and $S_n$ again converged to unity pretty convincingly. But I can't determine the needed conditions and prove theoretically convergence to unity, when the $a_i$'s vary (deterministically or as random variables).

I notice that we can write

$$S_n = a_1 + (1-a_1)\Big[a_2 +(1-a_2)\big[a_3+(1-a_3)[\cdots(1-a_{n-1}) a_n]\big]\Big]$$

All these seem eerily familiar but I can't pin down where I have seen these expressions before...

QUESTION: What conditions are needed and how can we prove convergence to unity of the above infinite series?

Answer 1

Your sense of "eerie familiarity" with the sequence $S_n$ was justified - in fact, we have $S_n = -(1-a_1)(1-a_2)(1-a_3)\cdots(1-a_n) + 1$. Clearly as $n$ goes to infinity, under reasonable conditions we can expect the value of the product term in $S_n$ to tend to zero - and so you have your limit!