Convergent Series of Rectangles to Fill a Unit Square

iterations

4

numerator n

1

denominator d

3

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

r=n/d

(numerator

n

, denominator

d

) of the unshaded region of the square. The

th

k

shaded rectangle has area given recursively by

f

k

=

f

k-1

(1-r)

with

f

1

=r

, so that

f

k

=r

k-1

(1-r)

. The area

A

k

after

k

rectangles have been shaded is given recursively by

A

k

=

A

k-1

(1-r)+r

with

A

1

=r

. Thus

A

k

=1-

k

(1-r)

, which tends to 1 as

k∞

.