the.com/infinite series
Adding forever, yet sometimes landing on a perfectly ordinary number.
means A sum of infinitely many terms, added one after another forever — which can still settle on a single finite value, or can refuse to and blow up or wander.
from From Latin 'infinitus' (not finished, boundless) and 'series' (a row, chain, succession — from 'serere,' to join or link). The math sense of 'series' as a chain of added terms took shape as 17th- and 18th-century mathematicians like Newton, Leibniz, and the Bernoullis worked out how to add endless chains of numbers and, strangely, sometimes get a tidy finite answer — the formal rules for when that's allowed (convergence) were only nailed down rigorously later, in the 19th century.
- the trick
- terms must shrink faster than they add up
- classic case
- 1+1/2+1/4+... quietly equals 2
- cautionary tale
- 1-1+1-1+... depends on how you group it
- real use
- computers approximate sine and e this way