Does the sum of a constant over n converge?

Actually, it will never converge. Just because the individual terms approach zero, it doesn't mean that the whole thing approaches zero. This type of series is called a harmonic series, and there are several ways to prove that it diverges. If you know calculus, you can use the integral test and find that it approaches ln(infinity) which is infinity, or if you know p-series then it diverges by p-series.