Cahen's constant

The Cahen 's constant a of the French mathematician Eugène Cahen (1865 - ) named mathematical constant. It is a transcendental number and is defined as the limit of an alternating series of unit fractions.

Definition

The denominators of unit fractions are derived from the following members of the Sylvester - sequence expressed by recursively

Defined ( A000058 in OEIS sequence ). With this result, the Cahen 's constant by

Defined, that is, the Sylvester sequence is the Pierce - development of C. By the Leibniz criterion, the convergence of the series can be shown directly.

Properties

After merging of two members of the series is obtained a series whose terms are only positive unit fractions:

This representation also provides the greedy algorithm for unit fraction decomposition of C ( the denominator as the result of A123180 in OEIS ). The series converges rapidly because of the double exponential growth of the Sylvester sequence, each time you add taken summand quadrupled the number of valid points.

An approximate value for the Cahen 's constant

Eugène Cahen 1891 proved in an elementary way that C is irrational (this follows from the fact that the Pierce - development does not break off ). J. Les Davison and Jeffrey Shallit showed in 1991 that C is transcendent. Their proof shows more generally for all numbers whose continued fractions satisfy certain simple recursive education laws that they are transcendent. Especially for C, the continued fraction expansion is carried

Optionally, wherein the sequence recursively

Defined ( A006279 in OEIS sequence ).

Variations to the defining row of C is known to be

While and is still open, what can be said about ( the Sylvester sequence is in this case the angels development, so the limit is certainly irrational ).