TITLE: What is the smallest constant that has explicitly appeared in a published paper?
QUESTION [16 upvotes]: You can read a lot about what large number in mathematics are, like Skewes' number, Moser's number or even Graham's number. 
So just for the sake of non-discrimination, I ask, what is the smallest constant $\epsilon$ that has explicitly appeared in a published paper?
Comment: As usual let's assume $\epsilon >0$! And only numbers in the domain of real numbers are allowed (Thx to Holowitz).

REPLY [19 votes]: The article "Strange Series and High Precision Fraud", J. M. Borwein and P. B. Borwein, The American Mathematical Monthly, Vol. 99, No. 7, (Aug-Sep, 1992), pp. 622-640, contains some interesting small error terms. In particular on p.639: $$\left|\sqrt\pi-\left(\frac{1}{10^5}\sum_{n=-\infty}^{\infty}e^{-n^2/10^{10}}\right)\right|\leq 10^{-4.2\cdot 10^{10}} $$