Absolute convergence

The central fact is that absolute convergence implies convergence. If \(\sum|a_n|\) converges, its partial sums form a Cauchy sequence, and the triangle inequality \(|a_{m+1}+\dots+a_n|\le|a_{m+1}|+\dots+|a_n|\) makes the partial sums of \(\sum a_n\) Cauchy as well. By the completeness of \(\mathbb R\), the series converges, and moreover \(\left|\sum a_n\right|\le\sum|a_n|\).

The converse is false. The alternating harmonic series \(\sum(-1)^{n+1}/n\) converges (to \(\ln 2\)), but \(\sum 1/n\) diverges. A series that converges without converging absolutely is called conditionally convergent; its value depends on a delicate cancellation between positive and negative terms.

To decide absolute convergence, apply to \(\sum|a_n|\) any test for series of nonnegative terms, above all the comparison test: if \(|a_n|\le b_n\) and \(\sum b_n\) converges, then \(\sum a_n\) converges absolutely. Thus \(\sum \sin(n)/n^2\) converges absolutely because \(|\sin n|/n^2\le1/n^2\).

The difference between the two kinds of convergence runs deep. In an absolutely convergent series the terms can be rearranged without changing the sum; in a conditionally convergent one, Riemann's theorem says a suitable rearrangement produces any prescribed sum, or even divergence. This is why absolute convergence is the natural notion when manipulating series.