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# Cahen’s constant

Whereas a simple addition of unit fractions with the terms of Sylvester’s sequence as denominators gives as a result the integer 1, an alternating sum

$\sum_{{i=0}}^{\infty}\frac{(-1)^{i}}{a_{i}-1}$ |

(where $a_{i}$ is the $i$th term of Sylvester’s sequence) gives the transcendental number known as *Cahen’s constant* (after Eugène Cahen) with an approximate decimal value of 0.643410546288338026182254307757564763286587860268239505987 (see A118227 in Sloane’s OEIS). Alternatively, we can express Cahen’s constant as

$\sum_{{j=0}}^{\infty}\frac{1}{a_{{2j}}}.$ |

The recurrence relation $b_{{n+2}}={b_{n}}^{2}b_{{n+1}}+b_{n}$ gives us the terms for the continued fraction representation of this constant:

$1+\frac{1}{{b_{0}}^{2}+\frac{1}{{b_{1}}^{2}+\frac{1}{{b_{3}}^{2}+\,\cdots}}}$ |

Synonym:

Cahen constant

Major Section:

Reference

Type of Math Object:

Definition

Parent:

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## Comments

## Continued fraction problem continues...

Continued fraction looks good in "page images" but not so good in "HTML with images." There's at least one other, older, PM entry with this problem.

## Re: Continued fraction problem continues...

I submitted a correction ages ago to "top 10 coolest numbers"

http://planetmath.org/?op=getobj&from=objects&id=7563

which was for the same problem. The owner fixed it by changing cfrac to sfrac. This is the same as is used in

http://planetmath.org/encyclopedia/ContinuedFraction.html

For some reason PM seems to like one but not the other

## Re: Continued fraction problem continues...

I looked at both of those. I tried simply changing the cfracs to sfracts, but I keep getting rendering errors and the oh-so-very helpful "view source annotated with errors" where the whole thing is red.