From Sequences to Series

Calculus 2 · Axiom Academy

A runner covers half the remaining distance each step — watch infinitely many shrinking pieces add up to a finite total. You're running to a finish line at 1 , but each step covers only half of the distance still left — 1/2 , then 1/4 , then 1/8 … Do you ever arrive? Hit Go and take the steps one at a time. Each step closes half of whatever gap is left — the pieces 1/2, 1/4, 1/8, … pile up and the runner homes in on the finish. Add up the first few step-sizes and you get a partial sum . Drag in more terms and watch the partial sums S₁, S₂, S₃, … climb toward the limit — close in fast, but never crossing 1. Same idea, out in the world: drop a ball and it returns to a fraction r of its height each bounce. Drag the bounciness and watch its total up-and-down travel — another shrinking geometric series — settle on a finite number a ÷ (1 − r). Zeno was wrong about the impossible: infinitely many shrinking pieces really do add to a finite whole. When each piece is a fixed fraction r of the last and |r| < 1 , the series converges — and the sum is exactly a ÷ (1 − r) . That one fact runs repeated drug doses, bouncing balls, perpetuity payouts, and every infinite decimal like 0.999… = 1.

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