How Can Zeno's Paradox Be Disproved Using Infinite Series?

[bjon-07]

Please help...how to disprove Zenos Paradox

I have a physics test tomorrow and I am sure my prof. will ask us how to disprove Zenos Parsox (about the turtle and the hare). I know I learned how to do this in cacl 2 using infinite series but I don't remember how to do that now. If some one can give me the equations, or point me in the righ diretion I would greatly appreate it. I can solve the equation my self if it is given to me, just don't remember how to set it up.

Thank you for your time

 

[HallsofIvy]

How to disprove it? Bring a turtle and a hare to class with you and let them go!
(Very 'Zen')

The point of Zeno's paradox was the hare has to cross an infinite number of points in order to catch up to the turtle and that (he claimed) must require an infinite amount of time. (That's the logical error- you do not have to take an infinite time to cross an infinite number of points.)

You, I believe, are thinking about the fact that the infinite sum:
1/2+ 1/4+ 1/8+ 1/16+ ... +1/2ⁿ+ ...

has the finite sum, 1.

I don't really consider that to be a "disproof" of Zeno's paradox (the one about Achilles and the turtle, not the "tortoise and the hare"- that comes from Aesop.)
but an easy way to prove it is by noting that, if S= 1/2+ 1/4+ ...
then S- 1/2= 1/4+ 1/8+ ...= (1/2)(1/2+ 1/4+ ...)= (1/2)S.

Now solve for S (and hope that your teacher doesn't call on the legitimacy of all those operations on infinite series).

Better would be to look at the finite sums: if S= 1/2+ 1/4+ ...+ 1/n²
then S- 1/2= 1/4+ 1/8+ ...+ 1/2ⁿ= (1/2)(1/2+ 1/4+ ...+ 1/2^n-1)

Now, unfortunately, that last sum is not the same as S. But we can make it the same by adding and subtracting 1/2ⁿ:
S- 1/2= (1/2)(1/2+ 1/4+ ...+ 1/2^n-1+ 1/2ⁿ- 1/2ⁿ
S- 1/2= (1/2)(1/2+ 1/4+ ...+ 1/2^n-1+ 1/2ⁿ)- 1/2ⁿ⁺¹
S- 1/2= (1/2)S- 1/2ⁿ⁺¹.

Now solve for S and then see what happens to that as n goes to infinity.

 

[arildno]

What's the difference between a tortoise and a turtle?
(PS: I know the difference between a hare and Achilles)

 

[HallsofIvy]

I'm sure Achilles is happy to hear that!

Turtles live all their lives in water. Tortoises typically live on land.

(In parts of the southern United States, tortoises are often called "gophers". I've heard that was because a person wanting to make turtle soup caught one and said "It's not a turtle but it'll gofur one!")

Now: what's the difference between a hare and a rabbit?

 

[bjon-07]

What's the difference between a tortoise and a turtle?
(PS: I know the difference between a hare and Achilles)

I am lazzy lazzy man, and typing tortoise is to much work.

 

[HallsofIvy]

But you can put in extra z's??

(Oh, wait, I just got it "extra z's"! Hah,Hah!)

(This thread is getting way too silly.)

 

[Bladibla]

Now: what's the difference between a hare and a rabbit?

http://www.madsci.org/posts/archives/dec99/946002744.Zo.r.html

 

[Locrian]

You should just tell him that it doesn't need disproving, because it was never shown to have any physical predictive power to begin with.

 

[drcrabs]

It is Time

Personally, i think that this theory is a waste of time. The whole fact that you took the time to prove time did not exist confuses me. I get where your coming from but, and i think i speak for everyone here, who cares!

 

[HallsofIvy]

Personally, i think that this theory is a waste of time. The whole fact that you took the time to prove time did not exist confuses me. I get where your coming from but, and i think i speak for everyone here, who cares!

Interesting. I went back and re-read this thread and found NO mention of prove time does not exist. What ARE you talking about? (And you think you speak for everyone here?)

 

[Undomiel]

What's the difference between a tortoise and a turtle?
(PS: I know the difference between a hare and Achilles)

Turtles live all their lives in water. Tortoises typically live on land.

Also - turtles have flippers, and tortoises have toed feet

Thanks for your explanation of solving (one of) Zeno's Paradox(es), HallsofIvy ... I recently had one of my favourite Physics professors explain this to me personally, very similarly to your explanation ... the only problem is, I understand yours, and my Professor's, explanations ... but I am not capable of explaining it to anyone else. I suppose this means I have not really taken it in well enough yet