The harmonic series

The harmonic series is
H = 1 + 1/2 + 1/3 + 1/4 + ....
Does its sequence of partial sums converge? Use the calculator below to investigate this question. Then try to formulate an argument which establishes you conclusion beyond any reasonable doubt. Such an argument was given by Nicolas Oresme (1323 - 1382 A.D.). [ Don't peek at that last link until you've thought about this. ]
     Harmonic series calculator
Number of terms: Partial sum: