$i+=1/(($_+=1e-7)**2+1)until$_>1;print$i/1e7*4
#23456789_123456789_123456789_123456789_123456
#        1         2         3         4
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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
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(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.)