Infinite Geometric Series

Pre-Calculus · Axiom Academy

LESSON Infinite Geometric Series When does adding forever land on a finite number — and what number is it? 1. A Sum That Sneaks Up on a Number We already know how to add a finite geometric series. But what if the terms never stop? Keep a running total — the partial sum S_n — and watch where it goes. For each new term is half the gap that is left, so the total climbs toward 1 and never overshoots . An infinite geometric series: keep adding a_1 r^n forever Whether the partial sums settle down comes entirely from the common ratio r . The n -th term is a_1 r^n , so the whole question reduces to one thing: does r^n shrink to 0 , or not? Watch r^n for two ratios at once. If the terms don't vanish, so the series diverges — there is no finite sum. has partial sums — it oscillates and has no limit. Only here do the terms shrink fast enough for the partial sums to reach a finite limit. 3. The Sum Formula, and Where It Comes From Start from the finite-sum formula and let n run to infinity. The finite sum of the first n terms is . The animation tracks r^n (orange) and S_n (blue) together: as r^n melts to 0 , the 1-r^n becomes just 1 , and S_n glides up to its limit. The derivation is one clean limit: Because |r| < 1 , repeatedly multiplying by r drives . That single fact is the whole engine of the formula. With : — exactly the limit we watched in Step 1.

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