1/2 + 1/4 + 1/8 + ... = 1: filling a square

A unit square is progressively filled by repeatedly halving the remaining area, generating the series 1/2 + 1/4 + 1/8 + ... The animation tracks partial sums 1/2, 3/4, 7/8, ... visually approaching 1 while the leftover unfilled region shrinks toward zero. This geometric picture motivates the general formula for an infinite geometric series, sum r^n = 1/(1-r), showing convergence rather than just stating it. Useful for students meeting infinite series and geometric sums for the first time.

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The prompt that made it

A unit square is filled by halving what remains: 1/2, 1/4, 1/8, ... The partial sums 1/2, 3/4, 7/8, ... approach 1 and the leftover shrinks to zero; the general formula sum r^n = 1/(1-r) closes.

Make your own version

Make the next one in this series

• Show the same halving process but with ratio r = 1/3 instead of 1/2.
• Animate why the geometric series formula fails or diverges when r is greater than or equal to 1.
• Compare partial sums of a geometric series to a harmonic series to show divergence versus convergence.

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