Tell me, how dose calculus disprove Zeno's paradox? It looks pretty solid to me.?

Question

I never got in to math in school, but as An adult I started to learn it agin and found a paradox involving the number line. I later found out about Zeno's paradox (Its the same as the one I found on the number line) People say Netwon solved it with his Calculus but How?

Answer 1

Imagine you are walking along the number line from 0 to 1 and that you are walking at 1 unit/second, i.e., it should take you 1 second to walk the total distance.

Zeno's paradox says that you shouldn't be able to reach 1, since before getting to 1, you have to first reach the point 1/2, and then the point 3/4, and then the point 7/8, etc., and there are infinitely many points that you have to go through before you reach 1. Surely, it should take an infinite amount of time to pass through these infinitely many points, right?

In fact, it doesn't. First, the amount of time it takes to go from 0 to 1/2 is 1/2 second. Then, to go the additional 1/4 unit of distance to 3/4, it will take 1/4 second. To go the additional 1/8 unit of distance to 7/8, it will take 1/8 second. This continues ad infinitum.

Thus, the total time that has elapsed during the walk to the point 7/8 is (in seconds)
1/2 + 1/4 + 1/8.
In general, you can walk "almost" all the way to 1 by going through n steps of this sort, and the time it will take is
1/2 + 1/4 + 1/8 + ... + 1/2^n

So far, no calculus. Now the reason the paradox seems sound is that the time it takes to go n steps is a sum of n numbers, as in the sum above. You would think that as you sum more and more numbers, the total time will keep increasing and increasing to infinity.

This is where calculus intervenes. In fact, these sums NEVER get bigger than 1. Try it on a calculator. Add up 1/2 + 1/4 + 1/8 + 1/16 + ... + 1/65536, where 65536=2^16 so this is the sum for the first 16 steps of your journey. It is very slightly below 1. In fact, as you add up more and more terms, these "partial" sums of the entire time your walk takes get closer and closer to 1. In calculus, we say that these sums are approaching a LIMIT. This is the fundamental concept upon which all of calculus is based. In general, if some quantity gets closer and closer to some value under some process, we say that the limit of the quantity is that value.

With our example, although it appears that the sum of the times it takes to walk each piece of the journey should be infinite, because there are infinitely many terms, we find that this argument is fallacious because if you actually try adding up a whole bunch of terms, it never exceeds 1. This is quite a surprising result, no? Because of this, we say that the infinite sum

1/2 + 1/4 + 1/8 + ...

with infinitely many terms converges to 1, i.e., we assign a value to what these terms add up to, because if you try adding as many terms as you please, you will find that it just keeps getting closer and closer to 1.

Answer 2

The long and short of it is that the paradoxes can be described by various geometric series. Depending on the problem and with certain methods of calculus, we see that these infinite series do converge to a finite number.

Answer 3

I am not doing your homework for you thanks for the points