Compute

Notice that the first and last terms = 2008

And notice that the second term and the next-to-the-last term  = -2 - 2206 = -2008

And there are 1002 pair-wise additions like this that will sum to "0"

That leaves three "middle" terms, 1003 - 1004 + 1005 = 1004....and that's the sum

Do this as 2 seperate APs and then add the sums together.