Cynthia Lanius

Explanation of One Half

Visualizing An Infinite Series


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

Explanation

    Do you see that every colored section has a corresponding uncolored one? That means that the sum of all the colored sections is one half.

    But what about the small square in the middle of the figure. Think of that square as being a small copy of the large square. If you magnified that square, it would be colored in the same pattern as the larger one. If you repeated the coloring process infinitely many times, exactly one half that square would be colored.

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