Go to Fun_Math Content Table Sums of Integers and Series
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)
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
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
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Last Updated July 9-th, 2006
Copyright 2006 Takaya Iwamoto All rights reserved.
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