From infinite product to infinite sum and a needed expression

Question

We have some function $f_n$ where $f_{\infty}=0$. Now we have a recursive function $t$ where $$t_{n+1}=\Big(1-f_n\Big)t_n$$

Does this converge?

Determine $t_{\infty}$

Answer 1

For all $n$ we clearly have :

$$1 \geq \left(1-\left ( \frac{3}{4}\right)^{2^np} \right )$$ Hence for all $n$ we have :

$$t_{n+1} \leq t_n$$

Because $t_n \geq 0$ for all $n$ and that the sequence $(t_n)_{n \in \mathbb{N}}$ is decreasing then it converges.