1 + x + x^2 + x^3 = 0
1 + x + x^2(1+x) = 0
(1 + x) ( 1 + x^2 ) = 0
解得x=-1
所以
x + x^2 + x^3 + ...+ x^1999 + x^2000 + x^2001
= -1 + 1 - 1 + ...-1 + 1 - 1
= -1