Infinite Geometric Series
On this web page, you will have a chance to study the
properties of infinite geometric series. This exploration will
reinforce the concepts in Lesson 9.2 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.
Investigate
- Press Show 1/2. How much of the square is
left over?
- Now press Show 1/4. What sum shows how much
of the square is colored? What fraction tells how much is left over?
- Now press Show 1/8. What sum shows how much
of the square is colored? What fraction tells how much is left over?
- Continue. How much of the square is left over after
you have pressed up through Show 1/m?
- What would the result be if you could sum all
infinitely many terms in the series
 ? Explain.
- 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, tell the
total amount of pie eaten so far.
c. If Jack lives forever, how much of the pie will he eat?
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