Absolute and conditional convergence

Level UniversityDifficulty ★★★★★Concept⌖ Open in the map

What is it?

$\sum a_k$ converges absolutely if $\sum |a_k|$ does. Absolutely convergent series can be reordered freely; conditionally convergent ones cannot — Riemann showed a rearrangement can make them sum to any number.

Formulas

\sum |a_k| < \infty \implies \sum a_{\pi(k)} = \sum a_k \ \ \forall\,\pi

Where it shows up in computing

Floating point (IEEE 754)★★★★★frequentScientific computing and algorithms
Float addition is not associative; parallel reductions reorder terms and give run-to-run differences — order matters.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

λ Scientific computing and algorithms

Floating point (IEEE 754)★★★★★
Floating point (IEEE 754)→Scientific computing★★★★★

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.