matematicas visuales visual math
Sum of a Geometric Series

We have already seen, studying the geometric series of ratio 1/4, that a geometric series

is a convergent series when the ratio is less than 1. Its sum is:

A simple case is when the ratio is

Then, the geometric series we want to add can be represented in this way:

Representation of several terms of the geometric series of ratio 1/2 | matematicasvisuales
The sum of the geometric series of ratio 1/2 is 1 | matematicasvisuales

The sum of the geometric series of ratio 1/2 is:

In this visualization we have used a rectangle with a particular shape: when we cut it in half we get two rectangles with the same shape. This aspect ratio is used, for example, in the international papel size standard, ISO 216 (based on the German DIN, for example, DIN A4).

rectangle aspect ratio, DINA4 | matematicasvisuales

The aspect ratio of these rectangles is:

In the animation, these rectangles follow a kind of spiral around a point. ¿Can you think what are its coordinates? The result is related with another geometric series: the geometric series of ratio 1/4.

The center is related with the geometric series of ratio 1/4 | matematicasvisuales


Geometric sequence
Geometric sequences graphic representations. Sum of terms of a geometric sequence and geometric series.
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The integral of power functions was know by Cavalieri from n=1 to n=9. Fermat was able to solve this problem using geometric progressions.
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Dilative rotation
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Equiangular spiral
In an equiangular spiral the angle between the position vector and the tangent is constant.
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Complex Geometric Sequence
From a complex number we can obtain a geometric progression obtaining the powers of natural exponent (multiplying successively)