# Convergent Series of Rectangles to Fill a Unit Square

Convergent Series of Rectangles to Fill a Unit Square

This Demonstration shows graphically how the sum of a certain infinite series converges to 1. Blue rectangles are successively added inside a square of area 1. Each iteration shades a fraction (numerator , denominator ) of the unshaded region of the square. The shaded rectangle has area given recursively by =(1-r) with =r, so that =r. The area after rectangles have been shaded is given recursively by =(1-r)+r with =r. Thus =1-, which tends to 1 as .

r=n/d

n

d

th

k

f

k

f

k-1

f

1

f

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k-1

(1-r)

A

k

k

A

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A

k-1

A

1

A

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k

(1-r)

k∞