absolute convergence

Absolute convergence is the strong, sturdy kind of convergence: not only does the series converge, but it would still converge even if you stripped away all the signs and made every term positive. There is no delicate cancellation propping it up — the terms are genuinely small enough on their own.

Precisely, a series sum a_n converges absolutely if the series of absolute values sum |a_n| converges. The key theorem is that absolute convergence implies (ordinary) convergence: if sum |a_n| < infinity, then sum a_n converges as well. The proof uses completeness via the Cauchy criterion, since |a_{m+1} + ... + a_n| <= |a_{m+1}| + ... + |a_n| makes the tails of sum a_n small whenever those of sum |a_n| are.

Absolutely convergent series are robust in ways ordinary convergent ones are not: their terms may be rearranged in any order without changing the sum, and two of them may be multiplied via the Cauchy product with a guaranteed valid result. This robustness is exactly what conditional convergence lacks, which is why the absolute–conditional distinction is fundamental.