Mathematica, 27 chars (arguably as low as 26, or as high as 33)
NSum[8/i/(i+2),{i,1,9^9,4}]
If you remove the initial "N" then it returns the answer as a (huge) fraction.
If it's cheating that Mathematica doesn't need a print statement to output its result then prepend "Print@" for a total of 33 chars.
NB:
If it's cheating to hardcode the number of terms, then I don't think any answer has yet gotten this right. Checking when the current term is below some threshold is no better than hardcoding the number of terms. Just because the current term is only changing the 6th or 7th digit doesn't mean that the sum of enough subsequent terms won't change the 5th digit.
