#1. "RE: D3000 not working with AF S DX 55-200 f/4-5.6 ED" In response to Reply # 0
Welcome to Nikonians! Sorry to hear of your problem. First check the electrical contacts on the lens and make sure they all project out the same distance. They are spring loaded and can get jammed in. Second, make sure the lens is fully seated and locked in the camera mount. After it clicks gently rotate the lens back and fourth to verify it is locked in the mount. If the problem persists, the electrical contacts on the lens or the body might be oxidized. Clean them with a soft clean cloth or better yet Deoxit available at Radio Shack. Good Luck and Enjoy your Nikons!
#6. "RE: D3000 not working with AF S DX 55-200 f/4-5.6 ED" In response to Reply # 4
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!