$$A=a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{a_4 + ....}}},$$

where $a_n = f(n), \; n \in \mathbb{Z}^+$.

(?) Is the fraction above always convergent if $a_n \ge 1$?

(?) What about $ 0 < a_n < 1$ ?

(?) $a_n < 0$?

What is simplest way to calculate the sum of the following?

$$S_1=1 + \cfrac{1}{2 + \cfrac{1}{3 + \cfrac{1}{4 + ....}}};$$

$$S_2=1 + \cfrac{1}{2 + \cfrac{1}{4 + \cfrac{1}{8 + ....}}}.$$