Can Continued Fractions be Evaluated?

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SUMMARY

The discussion centers on the evaluation of a complex continued fraction expressed in infix and postfix notation. The fraction converges to approximately 0.38, and its evaluation relates to Bessel functions, specifically BesselI(0,2) and BesselI(1,2). The sequence is identified as the inverse of Sloane A052119, which can be found at the provided link. The conversation also touches on the challenges of evaluating arbitrary continued fractions and the role of moderators in forum discussions.

PREREQUISITES
  • Understanding of continued fractions and their convergence properties.
  • Familiarity with Bessel functions, particularly BesselI(0,2) and BesselI(1,2).
  • Knowledge of Sloane's OEIS (Online Encyclopedia of Integer Sequences) and its usage.
  • Basic skills in mathematical notation, including infix and postfix expressions.
NEXT STEPS
  • Research the properties and applications of Bessel functions in mathematical analysis.
  • Explore the concept of continued fractions and their convergence criteria in depth.
  • Learn how to utilize the Online Encyclopedia of Integer Sequences (OEIS) for mathematical research.
  • Investigate methods for evaluating complex continued fractions and their significance in pure mathematics.
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Mathematicians, students studying advanced calculus or analysis, and anyone interested in the evaluation of continued fractions and Bessel functions.

csprof2000
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If anybody can evaluate this, please let me know...

Infix notation:
1/(2+3/(4+5/(6+7/(8+9/(10+11/(...))))))

Postfix notation:
1 2 3 4 5 6 ... n ... / + / + / + / + ... / + ...
 
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I can't evaluate it but I can look it up. It's BesselI(0,2)/BesselI(1,2). The inverse of Sloane A052119. http://www.research.att.com/~njas/sequences/A052119
 
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Dick said:
I can't evaluate it but I can look it up. It's BesselI(0,2)/BesselI(1,2). The inverse of Sloane A052119. http://www.research.att.com/~njas/sequences/A052119

Ooops. I was reading it as a simple continued fraction, and it's not, sorry.
 
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Why was this moved to homework help? This is no homework problem. I challenge whoever dared move this to this forum to provide the answer or relinquish their moderation powers. Silly mods.
 
The mods are just trying to make this a livable place. Sometimes they make mistakes. A moderator that makes occasional mistakes in relocating threads is a lot better than no moderation. Trust me. But anyway, do you have a good reason for thinking there is an evaluation and why do you want to know? Just curious.
 
If you like, one of those "silly mods" could just delete this for you.
 
Whoa guys, no offense meant. Sheesh, serious mods.

Anywho, the reason I was wondering about this was that a student asked about it in my class. Apparently this thing does converge to around ~0.38, and I was just wondering if a more "pure" mathematician could help me out.
 
If you think about it, it would be just as hard to 'evaluate' (express in terms of elementary functions) an arbitrary continued fraction as an arbitrary string of digits. About all you can do is see if somebody has stumbled across it somehow. I took 1+1/(2+1/(3+1/(4+1/(5+1/... and evaluated it to a reasonable number of significant digits and hunted for it in Sloane's list. Was pretty happy to find something until I realized I'd gotten your fraction wrong. I don't find a hit for your form.
 

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