Does 1/2 plus 1/4 plus 1/8 and so an ad infinitum eventually equal one?
2011-10-29 1:31 am
For example, lets use distance. So there's twelve inches of space, a foot. You move half of the last step you took, starting at 6 inches. You can move any amount, no matter how tiny. I think that eventually (that is stressed very much; I think it would be an extreme amount of time((if there was time in this... "Place")) before it reached there) it would reach one, or one foot in the example. Because if you think about it, it's always going. Each step is distance. Distance eventually adds up. Will it eventually reach one?
回答 (8)
2011-10-29 1:53 am
✔ 最佳答案
The infinite sum is the nature of Zeno's Paradox and does add to 1.It is not the sum of the terms 1/(2n +2) as can be seen when n=2, which produces the fraction 1/6, which is not part of the sum you (and Zeno) suggest.
Instead it is the convergent sum of the terms 1/2^n, where n starts with 1 (not zero).
The proof would take up too much space here but you can consult any advanced algebra chapter on infinite series for that.
2011-10-29 2:13 am
The replies are essentially correct - It is Zeno's paradox but the answer is quite simple:
Just because something is infinitely divisible does not mean that it is infinitely long.
This is a classic mis-use of calculus where the infinitely small step is 'dx' but you try to measure it in time T and not a small amount of time dT. dx / T has no meaning but dx / dT does have a meaning and will yield an answer - in your case - equal to one (1).
Just because something is infinitely divisible does not mean that it is infinitely long.
This is a classic mis-use of calculus where the infinitely small step is 'dx' but you try to measure it in time T and not a small amount of time dT. dx / T has no meaning but dx / dT does have a meaning and will yield an answer - in your case - equal to one (1).
2011-10-29 1:54 am
numbers are finite so yes(i assume, getting headache now), but its going to be a vast amount of steps, i am going to write a little visual basic program tomorrow and give it a try, very interesting question
me edit: yeah after a quick google(do not know why i cannot remember the zeno machine) sure i did one at school the memory soon ran out, yes it must be solvable if you have infinite computing time
me edit: yeah after a quick google(do not know why i cannot remember the zeno machine) sure i did one at school the memory soon ran out, yes it must be solvable if you have infinite computing time
2011-10-29 2:09 am
No, you never reach the end.
2011-10-29 1:39 am
The sum you mention, with general term 1/(2n+2), diverges.
2011-10-29 1:34 am
no way it will just become a more exact number.
收錄日期: 2021-04-20 21:39:38
原文連結 [永久失效]:
https://hk.answers.yahoo.com/question/index?qid=20111028173133AAEKcWf