Catalan's constant

Catalansche the constant commonly referred to, is a mathematical constant. It is the value of the series

So the value of the Dirichlet beta function at the location 2 The constant is named after Eugène Catalan. Your irrationality is suspected, but is still unproven. It is known that infinitely many of the numbers β (2k ), k = 1, 2, 3 ..., must be irrational, while at least one of β ( 2), β ( 4), β (6 ), β (8) β (10), β (12) and β (14).

History and Description

Catalan described this constant in a work of 1867 with G and gave numerous integral and series representations of it.

Value

An approximate value is

Currently (April 16, 2009) are known 31.026.000.000 decimal digits.

Further illustrations

There is a rich abundance of other representations, a fraction of which is reproduced below:

Integral representations

Series representations

According to S. Ramanujan applies:

Another series contains the Riemann zeta function:

Very quickly converges following sum ( Alexandru Lupas 2000):

BBP -like rows

Man has long sought a BBP series. First, only very long specimens were found. Relatively short, the 9 -membered by Victor Adamchik (2007):