Solution
= Solution
A <series> $\sum_{n=1}^{\infty}u_n$ is a <convergent series> if its <partial sums> $S_N=\sum_{n=1}^Nu_n$ have a finite <limit> $S$ as $N\to\infty$. The number $S$ is the sum of the <series>. Explicitly, for every $\varepsilon>0$ there is an integer $N_0$ such that $|S_N-S|<\varepsilon$ whenever $N\geq N_0$. The condition $u_n\to0$ alone is insufficient: the <harmonic series> has terms tending to zero but diverges.