Alternating series

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

What is it?

$\sum (-1)^k b_k$ with $b_k$ decreasing to 0 converges (Leibniz), and the error of stopping at term $n$ is at most the first omitted term $b_{n+1}$ — a free error bound.

Formulas

\Big|S - \sum_{k=0}^{n}(-1)^k b_k\Big| \le b_{n+1}

Where it shows up in computing

Floating point (IEEE 754)★★★★★frequentScientific computing and algorithms
Summing alternating terms of large size cancels digits; evaluating $e^{-20}$ by its series in floating point gives garbage.

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.