Also, you need to realize that the reason a space elevator would work at all, is because the center of mass of the system is in a geostationary orbit. If you attach something massive to the tether and start cranking it up, the center of mass will be displaced downward, and thus the tether would want to move forward. The response is to crank something upward to keep the center of mass approximately the same; or to fire thrusters to adjust. Two words: Ballast Weight.
Increase it slightly so the centre of mass is always above geostationary orbit position. Tension on the cable is slightly greater, but it's always stable.
Increase it slightly so the centre of mass is always above geostationary orbit position. Tension on the cable is slightly greater, but it's always stable. Let's look at that. As the CM is above GEO, it will want to fall back. The tether has fixed length, so it will pull the CM downward. The net effect would most likely be that there's a slight catenary curve to the tether, with the CM at GEO, and you're back to the same situation. And that's just with the system itself -- once you add moving masses along the tether, the whole thing gets more dynamic.