Patterns to Infinity

Calculus 2 · Axiom Academy

Watch what a sequence does when it runs forever. Where do its terms end up — and what happens if you add them all together? A list of numbers that never ends A sequence is just an ordered list of numbers — but in Calculus 2 the lists go on forever. The whole game is asking what a list does in the long run: does it home in on a single value, or wander off? Get a feel for that one question and you've got the foundation for infinite series, Taylor series, and every "approximate it more and more precisely" method in the subject. Watch the simplest infinite sequence build itself: a n = 1/ n , so each term is one divided by its position — 1, then 1/2, then 1/3, then 1/4, and on forever. Press play and watch the terms stamp onto the number line. They never stop coming, yet they pile up tighter and tighter against a single spot: zero. The terms never reach zero — but they get arbitrarily close. That target they crowd toward is the limit of the sequence. Stacking the terms: a sum that fills up to 1 A sequence is a list; a series is what you get when you start adding the list up . Take 1/2 + 1/4 + 1/8 + 1/16 + … — each piece half the one before. Slide the handle to add more terms and watch the running total (the partial sum ) climb. It never overshoots, and with infinitely many terms it fills exactly up to 1 . Add infinitely many terms and the gap to 1 closes completely: 1/2 + 1/4 + 1/8 + ··· = 1. That value is the sum of the series .

This is the written version of the interactive lesson above. See the full Calculus 2 course.