Cynthia Lanius

Page 3

Visualizing An Infinite Series


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

Did you recognize the above relation from the drawings?If not, go back and take another look, or study my drawing.

Notice the two series that we have observed so far.

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

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

That makes me wonder if

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

Exploration 2

  1. Take a piece of square grid paper. Outline a 27 X 27 square. Divide the square into three rows, each 9 X 27. Color in the bottom row. Now divide the middle row into three equal columns and color in the right column. Now color in the bottom third of the middle column. Then divide the middle row above it into three equal columns and color in the right column. Continue this pattern.

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

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Copyright 1999-2008 Cynthia Lanius