Terms of a convergent series tend to zero
= Terms of a convergent series tend to zero
{title2=$\sum_{n=1}^{\infty}u_n\text{ converges}\ \Longrightarrow\ u_n\to0$}
If the <partial sums> $S_n=\sum_{j=1}^nu_j$ 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.