Question

The Greek philosopher, Zeno of Elea, presented a number of paradoxes that casued a great deal of conern to the mathematicians of his period. These paradoxes were important in the development of the notion of infinitesimals. One of the most famous is the race course paradox. It can be stated as follows: A runner can never reach the end of a race. As he runs the track, he must first run 1/2 the length of the track, then 1/2 the remaining distance. So that at this point, he has run 1/2 + 1/4 = 3/4 of the length of the track and 1/4 remains to be run. After running half the remaining distance, he finds that he still has 1/8 of the track left to run, and so on indefintely. a. Set-up the infinite series, in summation form, that gives the total distance covered by the runner trying to run a track that is 3 miles long. sum_{n=1}^{infty} b. What can be said about the convergence or divergence of this series? Series converges to Series diverges.

          The Greek philosopher, Zeno of Elea, presented a number of paradoxes that casued a great deal of conern to the mathematicians of his period. These paradoxes were important in the development of the notion of infinitesimals. One of the most famous is the race course paradox.
It can be stated as follows: A runner can never reach the end of a race. As he runs the track, he must first run 1/2 the length of the track, then 1/2 the remaining distance. So that at this point, he has run 1/2 + 1/4 = 3/4 of the length of the track and 1/4 remains to be run. After running half the remaining distance, he finds that he still has 1/8 of the track left to run, and so on indefintely.
a. Set-up the infinite series, in summation form, that gives the total distance covered by the runner trying to run a track that is 3 miles long.
sum_{n=1}^{infty}
b. What can be said about the convergence or divergence of this series?
Series converges to
Series diverges.

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The Greek philosopher, Zeno of Elea, presented a number of paradoxes that casued a great deal of conern to the mathematicians of his period. These paradoxes were important in the development of the notion of infinitesimals. One of the most famous is the race course paradox.
It can be stated as follows: A runner can never reach the end of a race. As he runs the track, he must first run 1/2 the length of the track, then 1/2 the remaining distance. So that at this point, he has run 1/2 + 1/4 = 3/4 of the length of the track and 1/4 remains to be run. After running half the remaining distance, he finds that he still has 1/8 of the track left to run, and so on indefintely.
a. Set-up the infinite series, in summation form, that gives the total distance covered by the runner trying to run a track that is 3 miles long.
sumn=1^infty
b. What can be said about the convergence or divergence of this series?
Series converges to
Series diverges.

Added by Ryan W.

Calculus: Early Transcendentals

James Stewart 8th Edition

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Solved by Expert William Semus

10/14/2023

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Step 1: The infinite series that gives the total distance covered by the runner can be set up as follows: ∑ (1/2)^n Show more…

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The Greek philosopher, Zeno of Elea, presented a number of paradoxes that caused a great deal of concern to the mathematicians of his period. These paradoxes were important in the development of the notion of infinitesimals. One of the most famous is the race course paradox. It can be stated as follows: A runner can never reach the end of a race. As he runs the track, he must first run the length of the track, then the remaining distance. So that at this point, he has run 1/2 of the length of the track and 1/2 remains to be run. After running half the remaining distance, he finds that he still has 1/4 of the track left to run, and so on indefinitely. a. Set up the infinite series, in summation form, that gives the total distance covered by the runner trying to run a track that is 3 miles long. b. What can be said about the convergence or divergence of this series?

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00:01 The greek philosopher zeno of elea presented a number of paradoxes that caused a great deal of concern to the mathematicians of his period.

00:10 These paradoxes were important in the development of the notion of infinitesimals.

00:16 One of the most famous is the racecourse paradox.

00:19 It can be stated as follows.

00:22 A runner can never reach the end of a race.

00:26 As he runs the track, he must first run half the length of the track, then half the remaining distance, so that at this point he has run a half plus a quarter, which is three quarters of the length of the track, and a quarter remains to be run.

00:44 After running half the remaining distance, he finds that he still has one eighth of the track left to run, and so on indefinitely.

00:52 So we're going to set up the infinite series in summation form that gives the total distance covered by the runner trying to run a track that is three miles long.

01:05 So first he has to run half the length of the track.

01:10 So it's a three mile long track, so he first has to run three halves of a mile.

01:18 Then he has to run half of the remaining distance.

01:22 Well, if he's already run three halves of a mile, three halves of a mile remain.

01:28 So he has to run half of that remaining distance, which is three quarters of a mile.

01:35 Now then he has to run half of the remaining distance again.

01:39 So there's three quarters remaining after he's run three halves plus three quarters, so he has to run another three eighths of a mile.

01:48 And this pattern continues, plus three sixteenths, plus dot dot dot, forever and ever...

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