Since it works perfectly fine with other lenses in P, S, & A modes, check the exposure settings (ISO, Shutter Speed, and Aperture (f/#) that the camera chooses in one of the Auto modes above, then change the exposure mode to Manual and set the same ISO, Shutter Speed, and Aperture settings that worked in Auto mode and capture an image. As long as the ambient conditions remain constant the exposure should be the same. In other words a correctly exposed image instead of a "black photo".
Since your 55-200mm works on your D40 there must be a problem with the electrical connection between the lens and your D3000. Good Luck and Enjoy your Nikons!