Cynthia Lanius

Page 2

Visualizing An Infinite Series

1/4 + (1/4)2 + (1/4)3 + . . . + (1/4)n + . . . = 1/3

Did you recognize the above relation from the drawings?If not, go back and take another look,or go to an explanation.

The above is an example of an infinite geometric series. Infinite, because it has an infinite number of terms. Geometric, because the ratio of any two consecutive terms is a constant.

It may be that you've never encountered the notion that an infinite series could have a finite sum. That is a very odd and intriguing idea. It seems that if you kept adding to the series long enough, it would eventually increase without bound, and the sum would go to infinity. But that is not the case.

    Exploration 1

  1. Take a piece of square grid paper. Outline a 32 X 32 square. Divide the square in half by drawing a diagonal. Color in half the square. Now divide the uncolored triangle into two congruent triangles and color in one of those triangles. Again, divide the uncolored triangle in half, and color in one. Think of doing this infinitely many times. Would you ever have to start coloring outside the original square? If not, it has a finite sum.

  2. Look at the figure you drew in #1,
    1/2 + (1/2)2 + (1/2)3 + (1/2)4 + . . . + (1/2)n + . . . = ?

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