Terms of a convergent series tend to zero (${\sum }_{n=1}^{\infty }{u}_n\text{ converges}⟹u_n\to 0$)

If the partial sums $S_n={\sum }_{j=1}^n{u}_j$ tend to a finite limit $S$, then $u_n=S_n-S_{n-1}\to S-S=0$. Thus failure of the terms to tend to zero proves divergence. The converse is false: the harmonic series has terms tending to zero but diverges.