 Solution

A convergent series ${\sum }_{n\ge 1}{u}_n$ has partial sums $S_N={\sum }_{n=1}^N{u}_n$ tending to a finite limit $S$. Explicitly, for every $\epsilon >0$ there is an integer $N_0$ such that $|S_N-S|<\epsilon$ whenever $N\ge N_0$. The convergence of the individual terms to zero is necessary but is not sufficient for convergence of the series.