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B what is the area enclosed by the n th curve in this

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(b) What is the area enclosed by the n th curve in this series? (Note that the n th curve in this series is obtained from the ( n - 1)st curve by adding a large number of small squares. For example, we obtain the second curve from the first one by adding four squares, each of which has side length 1 / 3 and thus area 1 / 9. So the area enclosed by the second curve is 1 + 4 / 9.) Your answer should involve summing a geometric series. Solution. To obtain the n th curve from the ( n - 1)st curve, note that the ( n - 1)st curve consists of 4 · 5 n - 2 line segments of length (1 / 3) n - 2 . On each of these line segments we will place a square of side length (1 / 3) n - 1 , and thus of area (1 / 9) n - 1 . So to get from the ( n - 1)st curve to the n th curve, we add squares having total area 4 · 5 n - 2 · (1 / 9) n - 1 ; we can rearrange this to get (4 / 5) · (5 / 9) n - 1 . The area enclosed by the n th curve in this series (for n 2) is therefore 1 + (4 / 5)(5 / 9) + (4 / 5)(5 / 9) 2 + · · · + (4 / 5)(5 / 9) n - 1 and all the terms but the first make up a geometric series. We can rewrite this as 1 + h (4 / 9) + (4 / 9)(5 / 9) + (4 / 9)(5 / 9) 2 + · · · + (4 / 9)(5 / 9) n - 2 i and recall that a + ar + . . . + ar k = a (1 - r k +1 ) / (1 - r ). With a = 4 / 9 , r = 5 / 9 , k = n - 2, the area enclosed is therefore 1 + 4 / 9 1 - (5 / 9) n - 1 1 - 5 / 9 ! This can be simplified to give 1 + (1 - (5 / 9) n - 1 ) and if n is very large this is very nearly 2. In fact, the area enclosed by this curve is very close to being a square of side length 2. 6. Start with the point 0. Flip a coin. If it comes up heads, move 2 / 3 of the way toward 1, if it comes up tails, move 2 / 3 of the way to 0 from wherever you are at the time. Repeat forever. The points you find are drawing a picture of the Cantor set. Verify that any point in the Cantor set will move to another point in the Cantor set under the coin-flipping-and-moving process. (B+S 6.3.26) Solution. Moving two-thirds of the way towards 1 takes the point x to (2 + x ) / 3. To see this, note that the distance from x to 1 is 1 - x ; the distance from (2 + x ) / 3 to 1 is 1 - 2 + x 3 = 1 - x 3 . 4
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Moving two-thirds of the way towards 0 takes the point x to x/ 3. So we need to show that if x is in the Cantor set, then so are (2 + x ) / 3 and x/ 3. But we did this in class, by looking at the ternary expansion of x . 5
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