Infinte Sum of Geometric Series

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Sums of Integers and Series

### In finite sum of geometric series

List of animations posted on this page.(Click the text to watch animation.)

(1/4) of Triangle

(1/4) of Square
#### I. Basic idea - Paper Folding idea by Sundara Row (Ref. 1)

A square paper is prepared. Make 4 creases,diagonally , horizontaly and vertically as shown in the first picture.

This will define 4 middle points. Using these 4 points,fold 4 triangles as shown by red .

Area covered by red is (1/2) of the square paper.

Apply the same process to the area not colored yet. This will give the second yellow area.

The yellow colored area is (1/2) of (1/2) , which is (1/2)^{2}.

Now the area covered by colors(red & yellow) is (1/2) + (1/2)^{2}

and the are not covered by colors is (1/4) = (1/2)^{2}
If this process repeated infinitely, then the colored area will eventually cover all the paper area.

This is equivalent to stating the following.

(1/2) + (1/2)^{2} + (1/2)^{3} + (1/2)^{4} + ... + (1/2)^{n}+ ... = 1

We know in general (when n is not 1)that infinte sum of geometric series of (1/N)^{k} type is 1/(n-1) .

What this example suggests is that if the process of increasing power can be done in the same pattern,

then it is possible to demonstrate a similar proof for other values of N.

**********************************gss_row.dwg** ********************************

Two interesting figures using triangle and square are shown in ref.2 and 3.(N = 4 case)

#### II. Triangle case

Infinite sum of geometric series with ratio = 1/4 is 1/3 as shown in the figure below.

If the area of the triangle is replaced by any arbitray angle, this same idea can be used to
do "angle trisection".

*******************gss_1_final.dwg** *****************

You can see the process in animation.

**To create this drawing and animation: **

** Load gss_1.lsp (load "gss_1")
**

Then from command line, type gss_1

#### III. Square case

Infinite sum of geometric series with ratio = 1/4 is 1/3 as shown in the figure below.

*******************gss_2_final.dwg** *****************

You can see the process in animation.animation

**To create this drawing and animation: **

** Load gss_2.lsp (load "gss_2")
**

Then from command line, type gss_2

#### References

- Row, T.Sundara : Geometric Exercises in Paper Folding. Dover Publishing, p.7-8 ,1966, original in 1905.

- Nelson,R.B. : Proofs Without Words: Exercises in Visual Thinking. MAA,p.121, 1993.

- Nelson,R.B. : Proofs Without Words II: More Exercises in Visual Thinking. MAA,p.111, 2000.

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Last Updated July 9-th, 2006

Copyright 2006 Takaya Iwamoto All rights reserved.
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