# Infinite Continued Fractions

#### tahirimanov19

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?

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

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

#### idontknow

$$\displaystyle s_{1}^{2} = 1+2(s_{1} -1)+(s_{1}-1)^{2}$$.

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#### tahirimanov19

$$\displaystyle s_{1}^{2} = 1+2(s_{1} -1)+(s_{1}-1)^{2}$$.
It is a tautology...

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