Next: 2.5 Area Under the Up: 2. Some Area Calculations Previous: 2.3 The Area Under   Index

# 2.4 Finite Geometric Series

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

 (2.20)

Thus
 (2.21)

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.

2.22   Theorem (Finite Geometric Series.) Let be a real number such that . Then for all
 (2.23)

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

 (2.24)

As a special case, we have

which agrees with equation (2.21).

2.25   Entertainment (Pine Tree Problem.) Let be the subset of sketched below:

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.

2.26   Exercise.
(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.)

2.27   Exercise. A Let

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

 (2.28)

i.e. they knew the formula for a finite geometric series when .

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].

2.29   Exercise. Explain why formula (2.28) is a special case of the formula for a finite geometric series.

Next: 2.5 Area Under the Up: 2. Some Area Calculations Previous: 2.3 The Area Under   Index
Ray Mayer 2007-09-07