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
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.
