Zeno of Elea

Exercises

1. Formulate a proof by contradiction of the statement that there is no single largest (real) number.
2. 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 assertion?

3. Use the geometric series formula to express these repeating decimals in fraction form:

(a) .9999... (You may be surprised by this answer!)
(b) .259259259...
(c) .571428571428...
(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 original number.)
(e) .124999999...
4. Find the sum, if one exists, for the series:

(a) 54 + 18 + 6 + 2 + 2/3 + ...
(b) 1 - 1/2 + 1/4 - 1/8 + 1/16 - 1/32 + ...
5. 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 as fast.

(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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