conditional convergence

Conditional convergence is convergence that lives entirely on cancellation. The series converges, but only because positive and negative terms keep almost cancelling each other; strip the signs away and the total area is infinite. Such convergence is delicate — it depends on the order in which the terms arrive.

Precisely, a series sum a_n converges conditionally if it converges but sum |a_n| diverges. The standard specimen is the alternating harmonic series 1 - 1/2 + 1/3 - 1/4 + ..., which converges to ln 2 while its absolute-value series, the harmonic series, diverges to infinity. The alternating series test certifies the convergence; the n-th-term and integral tests certify the absolute divergence.

The defining fragility is captured by Riemann's rearrangement theorem: the terms of a conditionally convergent real series can be reordered to sum to any prescribed real number, or to diverge. In a conditionally convergent series the positive terms alone sum to +infinity and the negative terms alone sum to -infinity, and it is the precise interleaving that produces the finite sum. Absolutely convergent series suffer none of this.