The Geometric Series

Calculus 2 · Axiom Academy

Why an infinite sum of shrinking terms can land on a single finite number — and exactly when it does. 1. What Is a Geometric Series? A geometric series has the form . You get each term by multiplying the previous one by a constant common ratio r . The first term is a , and we can write the whole thing compactly with sigma notation. First term a , ratio r — each step multiplies by r 2. Visual Proof: The Classic Example Take the series , where and . Start with a unit square and keep cutting off half of what's left. Watch the running total climb toward — but never past — the whole. Each colored block is one term of the series, and together the blocks fill the entire square. The infinite sum equals exactly 1 — which is what predicts. Now the algebra. Take the partial sum S_n , multiply it by r , and line the two rows up. Almost every term cancels — that's the trick that collapses the whole sum into a short closed form. Subtract: S_n - rS_n = a - ar^n (the middle terms cancel) Factor: S_n(1 - r) = a(1 - r^n) As , if then , so the a r^n term vanishes and the partial sum settles down: The series converges if and only if . Here three series with a = 1 race upward as we add term after term. The two with flatten onto their ceiling ; the one with never stops climbing. Terms shrink to zero, so the partial sums approach a finite limit . Terms don't shrink to zero, so the partial sums grow without bound or oscillate forever.

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