Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2001 / Paper 2 / 12 / b

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

The series $\sum u_n$ has absolute convergence if $\sum |u_n|$ is a convergent series, meaning its nonnegative partial sums have a finite limit. Absolute convergence implies convergence of the original real or complex series. The converse fails: the alternating harmonic series converges, while its absolute-value series is the divergent harmonic series.