Zeno of Elea
1 + 2/3 + 4/9 + 8/27 + ...
Formulate a proof by contradiction
of the statement that there is no single largest (real) number.
Consider the infinite series formed
by adding up the powers of 2/3:
Use your calculator to
compute the sequence of partial sums for this series out to the 10th power
of 2/3. Can you tell from this whether the partial sums are getting closer
to a fixed value? What is this number, and how can you be sure of your
Use the geometric series formula to
express these repeating decimals in fraction form:
(a) .9999... (You may be surprised
by this answer!)
(d) .3058888888... (Hint: First
multiply this number by 1000, then convert the decimal part of the new
number into fraction form. Deduce from this the fractional form of the
Find the sum, if one exists, for the
(a) 54 + 18 + 6 + 2 + 2/3 + ...
(b) 1 - 1/2 + 1/4 - 1/8 + 1/16
- 1/32 + ...
Let's build a more careful analysis
of the Achilles paradox. You may wish to consult the diagram
we proposed. At the start of the race, say Achilles gives Bob a quarter-mile
headstart. Assume that Stage 1 of the race depicted in the diagram occurs
one minute later, so that Achilles runs at a speed of a quarter mile per
minute (Wow, that's fast!) and that poor Bob runs at a speed one tenth
(a) How much ground will be covered
by the runners between Stages 1 and 2? between Stages 2 and 3? If we add
up the infinitely many distances covered between successive stages of the
race, what sum do we get, and what can we conclude from this about the
outcome of the race?
(b) How long will it take before
Achilles catches up with Bob?
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last modified 8/28/02
Copyright (c) 2000. Daniel