Cynthia Lanius

Visualizing An Infinite Series

(A)



1

How can we show a visual representation of an infinite series -- a mathematical concept that is often only treated symbolically?

Let's look at this isosceles trapezoid.


(B)



1/4

You can divide it into 4 smaller trapezoids, each with an area 1/4 the original.

We'll focus on the yellow trapezoid.


(C)




+ 1/42

Here is the original yellow trapezoid, which is 1/4 of the green trapezoid in (A), plus a smaller yellow trapezoid that is 1/4 of the green trapezoid in (B). The smaller yellow trapezoid is 1/4 of 1/4, or (1/4)2.

(Remember that to find 1/4 of 1/4 we multiply 1/4 * 1/4, and a number multiplied by itself is squared.)





+ (1/4)3

Now we have the original yellow trapezoid, 1/4 of (A), plus the smaller yellow trapezoid from B (1/4 of 1/4), plus an even smaller yellow trapezoid that is 1/4 of the green trapezoid in (C). That makes 1/4 of 1/4 of 1/4, or ...?   ...right, (1/4)3.

(Remember that a number multiplied by itself twice is cubed.)



+ ... + (1/4)n + ... = ?

This notation means that we will be repeating this process infinitely many times.

Look at the figure. If we repeat the process infinitely many times, what fraction of the total figure will be yellow? That is our answer.

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