Geometric series in Adobe Illustrator

There’s a nice feature in Adobe Illustrator that makes it easy to visualize exponential change.  Here’s a little example to show your students how

\displaystyle\sum_{i=1}^\infty (\frac12)^i = 1

Start by drawing something… in this case, an elf.  Say he has a height of one unit.  Select your drawing, right-click, and group it.

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Now hit up the Effects > Distort & Transform > Transform

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Here’s where it gets cool.  First click Preview, then put a number in the Copies box (I used 100 to approximate infinity).  This will create a bunch of copies of the original, but since they’re all the same size, you can’t see them.  Change the horizontal and vertical scales to 50%, and space out the elves by increasing the Move>Horizontal slider.  Drop the vertical slider so that the elves are on the same baseline.

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You can now see that each elf is half the height of the previous.  This is a geometric sequence.  To find the value of the series, we need to add the heights of the elves, which we can do by adjusting the horizontal and vertical sliders to stack the elves on top of each other (make sure to have students make a prediction first).

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All that’s left is to see how tall that stack of elves on top of the first one is.  Remember that the big elf is one unit, so let’s use him as the measuring stick.  Copy and paste the big elf, then turn off his transformation effect and drag him up to measure the series.

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Students should see that the sum of the heights of all the little elves is equal to 1 big elf, which is to say 1/2 + 1/4 + 1/8 + … = 1.