Am I correct in thinking string theory has an infinite number of terms so to prove finiteness to the first order means proving one (or the first) term to be finite? If so, then how can we ever prove an infinite number of terms? And what exactly does it mean to say, or prove, something is finite to the first (or second, or third, etc..) approximation? What about it is finite? And what makes up one term? Please try not to be too technical with your answers or I will be lost and have even more questions...I'm just a curious layman.