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What is the limit of the continued fraction $$\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{5+\cfrac{1}{7+\cfrac{1}{11+\cfrac{1}{13+\cdots}}}}}}\ ?$$

Is the limit algebraic, or expressible in terms of e or $\pi$? What is the fastest way to approximate the limit?

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Does the sequence $2,3,5,7,11,13,\dots$ consist of all prime numbers? –  Américo Tavares Sep 12 '11 at 11:26
    
@Angela Richardson: Are you sure that it is convergent? Also, can you really use the expression of continued fractions to express a complex number such as $i$? Maybe I am missing something here, but I am a little confused. Thanks for any explanation. –  awllower Sep 12 '11 at 11:29
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@awllower: it converges by Sleszynski-Pringsheim. –  J. M. Sep 12 '11 at 11:40
    
@awllower: It is convergent by the Śleszyńsky-Pringsheim's Theorem. The continued fraction $\mathbf{K}(a_{n}/b_{n})$ converges if for all $n$ $$\left\vert b_{n}\right\vert \geq \left\vert a_{n}\right\vert +1.$$ –  Américo Tavares Sep 12 '11 at 11:40
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Reference: Chapter I, Theorem 4.1 of Continued fractions with applications by Lisa Lorentzen and Haakon Waadeland. –  Américo Tavares Sep 12 '11 at 11:46
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1 Answer

up vote 21 down vote accepted

As I've noted in the comments, the Śleszyński-Pringsheim Theorem guarantees that this continued fraction converges to a finite value (since the prime numbers are always greater than or equal to $2$).

This arXiv preprint gives an approximate value $s$ for the continued fraction whose partial denominators are prime numbers:

$$s=0.43233208718590286890925379324199996370511089688\dots$$

which the author computed with PARI/GP. He notes that the Plouffe inverter does not at all recognize this constant.


Here's some Mathematica code for computing the constant to prec or so significant figures:

prec = 500;
y = N[2, prec + 5]; c = y; d = 0; k = 2;
While[True,
  p = Prime[k];
  c = p + 1/c; d = 1/(p + d);
  h = c*d; y *= h;
  If[Abs[h - 1] <= 10^(-prec), Break[]];
  k++];
N[1/y, prec]

which yields the result 0.43233208718590286890925379324199996370511089687765131032815206715855390511529588664247730234675307312901358874751711021925473474173059981681532525370102846860319246045704466728602248840679362020193843643798792955246786129609763893526940277522319731978458635595794036202066338633654544895108909659715862787332585763686200183679952128087865043794610126643260422526400822552675221511335417037835319471839444535578027072057047898703982387228841680143293913635134426277200532815721973910424896503203478507

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For purposes of computation, you can use Lentz-Thompson-Barnett. –  J. M. Sep 12 '11 at 11:51
    
The OEIS has it, though: oeis.org/A084255 –  Charles Sep 12 '11 at 12:53
    
@J.M. can you calculate this for twin prime pairs instead of the prime (To a certain extent)...as its irrationality will prove the twin prime conjecture –  Shivam Patel Feb 19 at 16:23
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