Geometric Series · Visual Proof

The Square Halving
Paradox

½ + ¼ + ⅛ + … = 1

Width: 1

Height: 1

— px

Division 0

JUMP TO

Cumulative Sum

∑ (1/2)ⁿ so far

0

Start — click animate

Step by Step

Why It Converges

Each half is divided again into two equal pieces. The new piece is always exactly half the remaining empty space.

No matter how many times you halve it, the pieces always fit perfectly inside the original square — nothing overflows.

The infinite sum equals exactly 1 — the whole square, never more.

Geometric Series · r = ½ Sum = a / (1 − r) = 1 / (1 − ½) = 2 … scaled to unit square → 1 ∞ steps · 1 square

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