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October 19th, 2019, 10:27 AM  #1 
Senior Member Joined: Mar 2015 From: Universe 2.71828i3.14159 Posts: 169 Thanks: 65 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? $$S_1=1 + \cfrac{1}{2 + \cfrac{1}{3 + \cfrac{1}{4 + ....}}};$$ $$S_2=1 + \cfrac{1}{2 + \cfrac{1}{4 + \cfrac{1}{8 + ....}}}.$$ 
October 19th, 2019, 11:55 AM  #2 
Senior Member Joined: Dec 2015 From: Earth Posts: 826 Thanks: 113 Math Focus: Elementary Math 
$\displaystyle s_{1}^{2} = 1+2(s_{1} 1)+(s_{1}1)^{2} $.
Last edited by idontknow; October 19th, 2019 at 12:14 PM. 
October 19th, 2019, 12:32 PM  #3 
Senior Member Joined: Mar 2015 From: Universe 2.71828i3.14159 Posts: 169 Thanks: 65 Math Focus: Area of Circle  
October 20th, 2019, 03:07 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 21,110 Thanks: 2326 
$\displaystyle S_1 = \frac{I_0(2)}{I_1(2)}$, where $I_n$ denotes the modified Bessel function of the first kind.


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continued, fractions, infinite 
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