A series is a limit of partial sums, which is different from the limit of its terms.
All three displayed term sequences approach zero. The harmonic partial sums nevertheless grow without bound. The telescoping sum is N/(N+1), and the geometric sum is 1−2⁻ᴺ, both approaching 1. This directly separates a necessary test from a sufficient one.
A nonzero term limit proves divergence, but a zero term limit does not prove convergence.