Infinite Patterns

Pre-Calculus · Axiom Academy

Some patterns, continued forever, pile up to infinity — and some settle on a single number. This unit is about telling them apart. What happens when a pattern never stops? Imagine stacking blocks in a pattern: 1, then 3, then 5, then 7… and never stopping. Add an infinite list of numbers and you might expect the total to blow up to infinity. Sometimes it does. But not always — and the surprise is that an endless sum can land on one exact number. Watch the most famous example. Start with half a bar, then add half of what's left, then half of that, forever: . Press play and watch each piece snap into place — the running total never reaches past 1, and it never falls short of it. A never-ending sum that lands on one number is a convergent series — the total converges to a limit of 1. First, the list itself: a sequence Before adding anything up, look at the list of numbers on its own. An ordered list that follows a rule — like 1, 3, 5, 7, 9, … from the rule — is a sequence , and each number in it is a term . Some sequences run off to infinity. Others settle down. Drag to add more terms of the sequence below and watch where they head. As , the terms get as close to 1 as you like — that value the sequence heads toward is its limit . Add them up: does it settle, or run away?

This is the written version of the interactive lesson above. See the full Pre-Calculus course.