Infinite Continued Fractions

Mar 2015
182
68
Universe 2.71828i3.14159
I need help with ICFs, specifically calculating sum with algebra and/or simple calculus.

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

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

$$S_2=1 + \cfrac{1}{2 + \cfrac{1}{4 + \cfrac{1}{8 + ....}}}.$$
 
Dec 2015
972
128
Earth
\(\displaystyle s_{1}^{2} = 1+2(s_{1} -1)+(s_{1}-1)^{2} \).
 
Last edited: