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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
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k-1
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1
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k-1
(1-r)
A
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A
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k-1
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1
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(1-r)
k∞