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Cahen's Constant

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Cahen's constant is defined by the convergent alternating series

C=sum_(k=0)^(infty)((-1)^k)/(a_k-1)
(1)
=0.64341054628...
(2)

(OEIS A118227), where a_k is the kth term of Sylvester's sequence, indexed by a_0=2 and a_(k+1)=a_k^2-a_k+1.

Cahen (1891, pp. 512-513) proved that C is an irrational number, and Davison and Shallit (1991) proved that it is a transcendental number.

See also

Sylvester's Sequence

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References

Cahen, E. "Note sur un développement des quantités numériques, qui présente quelque analogie avec celui en fractions continues." Nouv. Ann. Math. 10, 508-514, 1891. https://www.numdam.org/item/NAM_1891_3_10__508_0/.Davison, J. L. and Shallit, J. O. "Continued Fractions for Some Alternating Series." Monatsh. Math. 111, 119-126, 1991. https://doi.org/10.1007/BF01332350.Sloane, N. J. A. Sequence A118227 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Cahen's Constant

Cite this as:

Weisstein, Eric W. "Cahen's Constant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CahensConstant.html

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