Apéry's Constant Approximations

E. Pegg Jr. (pers. comm., Nov. 8, 2004) found an approximation to Apéry's constant  given by

(1)

which is good to 6 digits.

M. Hudson (pers. comm., Nov. 8, 2004) found the approximations

			(2)
			(3)
			(4)
			(5)
			(6)
			(7)

where  is the Euler-Mascheroni constant and  is the golden ratio, which are good to 5, 7, 7, 8, 11, and 12 digits, respectively.

A curious approximation to  is given by

(8)

where  is the Euler-Mascheroni constant, which is accurate to four digits (P. Galliani, pers. comm., April 19, 2002).

Lima (2009) found the approximation

(9)

where  is Catalan's constant, which is correct to 21 digits.

REFERENCES:

Lima, F. M. S. "A Simple Approximate Expression for the ApŽry's Constant Accurate to 21 Digits." http://arxiv.org/abs/0910.2684/. 14 Oct 2009.