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October 19th, 2019, 10:27 AM   #1
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Math Focus: Area of Circle
Infinite Continued Fractions

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?

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

$$S_2=1 + \cfrac{1}{2 + \cfrac{1}{4 + \cfrac{1}{8 + ....}}}.$$
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October 19th, 2019, 11:55 AM   #2
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$\displaystyle s_{1}^{2} = 1+2(s_{1} -1)+(s_{1}-1)^{2} $.

Last edited by idontknow; October 19th, 2019 at 12:14 PM.
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October 19th, 2019, 12:32 PM   #3
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Originally Posted by idontknow View Post
$\displaystyle s_{1}^{2} = 1+2(s_{1} -1)+(s_{1}-1)^{2} $.
It is a tautology...
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October 20th, 2019, 03:07 AM   #4
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$\displaystyle S_1 = \frac{I_0(2)}{I_1(2)}$, where $I_n$ denotes the modified Bessel function of the first kind.
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