For each in
let denote the box

and let

I want to find the area of . I have

Since the boxes intersect only along their boundaries, we have

Thus

You probably do not see any pattern in the numerators of these fractions, but in fact is given by a simple formula, which we will now derive.

Proof:
Let

Then

Subtract the second equation from the first to get

and thus

**Remark**: Theorem 2.22 is very important, and you
should
remember it. Some people find it easier to remember the proof than to remember
the
formula. It would be good to remember both.

If we let
in (2.23), then from equation (2.20)
we
obtain

As a special case, we have

which agrees with equation (2.21).

Here , , , and is the point where the line intersects the line . All of the points lie on the line , and all of the points lie on the line . All of the segments are horizontal, and all segments are parallel to . Show that the area of is . You will probably need to use the formula for a geometric series.

(a) Find the number

accurate to 8 decimal places.

(b) Find the number

accurate to 8 decimal places.

(You may use a calculator, but you can probably do this without using a calculator.)

Use the formula for a finite geometric series to get a simple formula for . What rational number should the infinite decimal represent? Note that

The Babylonians[45, page 77] knew that

Euclid knew a version of the formula for a finite geometric series in the case where is a positive integer.

Archimedes knew the sum of the finite geometric series when . The idea of Archimedes' proof is illustrated in the figure.

If the large square has side equal to , then

Hence

i.e.

For the details of Archimedes' argument see [2, pages 249-250].