When you're comparing the sums of your series and the sum of the harmonic series... you're already assuming both converge -- and in fact they both diverge.

The problem you point to later is much finer than that first problem : finding that one and not the first is very surprising!

PS: you'll probably find wikipedia's page on integral-series comparison (http://en.wikipedia.org/wiki/Integral_test_for_convergence) very helpful.