Absolutely convergent and conditionally convergent series, and the general starting index

Let $(a_k)$ be a sequence of reals, with series $\sum a_k$ and partial sums $s_n = \sum_{k<n} a_k$ as in Series, partial sums, convergence and the sum, divergence, and the tail series, and let $|x|$ be the absolute value (Absolute value in an ordered field).

Absolute convergence. The series $\sum a_k$ converges absolutely when the series $\sum |a_k|$ converges (Series, partial sums, convergence and the sum, divergence, and the tail series). Since $|a_k| \ge 0$ for every $k$ (Basic properties of the absolute value), this is a statement about a series of nonnegative terms.

Conditional convergence. The series $\sum a_k$ converges conditionally when it converges (Series, partial sums, convergence and the sum, divergence, and the tail series, Limits and Cauchy sequences of reals) and does not converge absolutely.

So a convergent series is exactly one of the two: absolutely convergent or conditionally convergent, according as $\sum |a_k|$ converges or not.

One implication is already proved, and is not reproved anywhere on this page. If $\sum |a_k|$ converges then $\sum a_k$ converges states that if $\sum |a_k|$ converges then $\sum a_k$ converges. That lemma was coined and proved on the previous page of this track, where the root and ratio tests need it; this page names it and builds on it. In particular an absolutely convergent series is a convergent series, so the two words above really do partition the convergent series, and "conditionally convergent" is not vacuous by accident: the alternating harmonic series is a witness, and the witness is exhibited in FALSE: every convergent series converges absolutely.

General starting index. Let $m \in \mathbb{N}$ and let $(a_k)_{k \ge m}$ be a family from $m$ (Series, partial sums, convergence and the sum, divergence, and the tail series). The series $\sum_{k \ge m} a_k$ converges absolutely when $\sum_{k \ge m} |a_k|$ converges, and converges conditionally when it converges and does not converge absolutely. By Series, partial sums, convergence and the sum, divergence, and the tail series both statements are the corresponding statements for the shifted sequence $j \mapsto a_{j+m}$, so nothing new is being defined and every result below transfers to a general starting index in the same way, exactly as If $\sum |a_k|$ converges then $\sum a_k$ converges already records for the one implication it proves.

• Absolute convergence is a condition on the terms, not on the sum. It says the series of absolute values converges, and it says nothing about the value of $\sum_{k=0}^{\infty} a_k$. The two sums are in general different, and no statement here identifies them.

• Why the distinction earns a page. Every result on this page separates the two classes. An absolutely convergent series may be reordered at will (Dirichlet's rearrangement theorem: an absolutely convergent series converges unconditionally, and every rearrangement of it has the same sum) and multiplied by another (Mertens' theorem: if $\sum a_k$ converges absolutely to $A$ and $\sum b_k$ converges to $B$, their Cauchy product converges to $AB$); a conditionally convergent one may be reordered to any sum whatever (The Riemann series theorem: a conditionally convergent real series has, for every $c \in \mathbb{R}$, a rearrangement with sum $c$, and rearrangements diverging to $+\infty$, to $-\infty$, and oscillating with any prescribed $\liminf \le \limsup$ in $\overline{\mathbb{R}}$). The difference is not one of degree.

• A series of nonnegative terms converges absolutely if it converges at all, since then $|a_k| = a_k$. So the distinction is invisible for the comparison, condensation, Raabe, Gauss and Kummer tests of the previous page, all of which assume terms of one sign. It is not invisible on that page as a whole: the root and ratio tests are stated for terms of arbitrary sign and reach convergence of $\sum a_k$ precisely through If $\sum |a_k|$ converges then $\sum a_k$ converges, which is where the word absolutely convergent is first used. What that page does not develop, and this one does, is everything that separates the two classes rather than the one implication those two tests need.