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Problems > The Maine Page > 2002

A Sequence of Circles

I have an infinite number of circles. The first one has radius r. The second has radius r/2. The third has radius r/4, and so on.

I also have an infinite number of squares, each of which has area equal to the area of the corresponding circle.

What is the sum of the perimeters of all my squares?

View the solution

A Sequence of Circles Solution

The area of the n^th circle is
π r²(

1

4

)^n-1

Since each square has the same area as the corresponding circle, and the area of a square is s²
s=

π

r(

1

2

)^n-1
And the perimeter is 4s, or
4s = 4

π

r(

1

2

)^n-1
Therefore, the sum of the perimeters is:

∞

Σ4

π

r(

1

2

)^n-1

n=1

Which is the same as
4

π

r (1 +

1

2

+

1

4

+

1

8

+...)
This equals 4

π

r x 2 = 8
π
r

X, Y, and Z

Fractional Infinite Series

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