jmc

In the middle, we have $\cdots +1010-1000+990-\cdots$. If we match up the back with the front, and then do the same for the rest, we get pairs with $2000$ and $-2000$, so these will cancel out. In the middle, we have $2000-1000$ which doesn't cancel, but gives us $1000\to \boxed{1000}$.