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Infinite Geometric Series

On this web page you will have a chance to study the properties of infinite geometric series. This exploration will help you visualize the concepts in Lesson 11.2 of  Discovering Advanced Algebra: An Investigative Approach.

An infinite geometric series is a geometric series with infinitely many terms. This web page gives an example of a convergent series, for which the sequence of partial sums approaches a long-run value as the number of terms increases.

Sketch

This sketch shows an empty square that you can gradually fill up by pressing the Show buttons.

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Investigate

  1. Press Show 1/2. How much of the square is left?
  2. Now press Show 1/4. What sum shows how much of the square is colored? What fraction indicates how much is left over?
  3. Now press Show 1/8. What sum shows how much of the square is colored? What fraction indicates how much is left over?
  4. How much of the square is left over after you have pressed up through Show 1/m?
  5. What would the result be if you could sum all the terms in the infinite series 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ...? Explain.
  6. Jack baked a pie and promptly ate one-half of it. Determined to make the pie last, he decided to eat only one-half of the pie that remained each day.
    a. Show the amount of pie eaten each day for the first seven days.

    b. For each of the first seven days, write the total amount of pie eaten so far.

    c. If Jack lives forever, how much of the pie will he eat?