If the increase on each level is greater than the decrease of the increase then it diverges (assuming that means it can reach infinity)

1 = 1

1 + 1/2 = 1.5
1.5-1 = .5, .5/1 = .5

1 + 1/2 + 1/3 = 1.833333
1.8333-1.5 = .33333, .3333/.5 = .66666

So, the decrease in the increase is actually going down each level.. so it will reach forever (if you did 1 +1/2 + 1/4 the decrease is constant)

Honestly, I'm making this all up as I go, so I might not be right.