Is a string taut if the tension is zero? I'm studying vertical circles and a particle on a string will remain in motion as long as the string is taut. However, what if there is 0 tension in the string? Is it still technically taut? 
 A: The tension being zero is right at the moment of transition between the string being taut and the string being slack; usually in "particle on a string" problems you have to find the minimum or maximum of some value, and usually the trick is to take the tension equal to zero as this is where the string is "barely" taut.
A: revised
If the particle is moving just fast enough such that the particle is moving in a circle, the string is fully expanded because the particle is still moving in a circle, even though the string doesn't need to pull on the particle to maintain that circular motion (in this case gravity is usually doing all the pulling necessary). Since the string is fully expanded at this point, in this sense it is still taught. However, most people would relate taught to the feeling of, e.g. a guitar string, where the string resists your pushing into it. In this circular motion case, such a string would still end up feeling like it was resisting your pushing into it, since in order to change the shape of the string you would necessarily have to also move the ball around. So, It would still feel taught.
If the string isn't moving fast enough to maintain circular motion at the top, and instead there is a brief period where the ball is falling, the tension would also be zero here and there would be extra curves in the string, and so you wouldn't have to move the ball in order to move the string around, so in this case it would not feel taught.
