Number Theory Number Theory Math Forum

 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
Quote:
 Originally Posted by idontknow $\displaystyle s_{1}^{2} = 1+2(s_{1} -1)+(s_{1}-1)^{2}$.
It is a tautology... 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. Thanks from idontknow Tags continued, fractions, infinite Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post USAMO Reaper Math 7 February 1st, 2015 07:09 AM dkotschessa Number Theory 10 October 5th, 2013 03:00 AM Rich B. Real Analysis 1 September 12th, 2013 10:41 PM mathbalarka Number Theory 3 May 24th, 2012 09:55 AM roadnottaken Number Theory 1 February 19th, 2007 09:54 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top       