$i+=1/(($_+=1e-7)**2+1)until$_>1;print$i/1e7*4
#23456789_123456789_123456789_123456789_123456
# 1 2 3 4
46 chars. (Changes the algorithm.) Change all 7s to a higher number for a better approximation and slower runtime, change it to a lower number for worse approximation and faster runtime.
update:
$i+=($_&1?4:-4)/($_*2-1)for 1..1e6;print$i
#23456789_123456789_123456789_123456789_12
# 1 2 3 4
(another, similar solution was there breifly: 44 chars.Pretty much the same algorithm Ovid gave. (The 4 is just factored out.))
42 chars. The same algorithm Ovid gave. (Change the 6 to a higher number for a better approximation, etc.)