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A line from Infinite Powers by Steven Strogatz

The limit of the infinite sum of time to reach a finite distance converges to a finite value and thus, ending Zeno's Paradox (at least in Physics).

Here's my question, what does it mean to finish an infinite number of steps in a finite time because clearly, it is assumed that the limit is reached.

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    $\begingroup$ This is either a mathematics or philosophy question depending on what spin you wanna put on it. I don't think it has much to do with physics. Mathematically, it's called a convergent series and you can look that up. Philosophically, I'mma head out :p $\endgroup$ Jun 23, 2021 at 12:06
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    $\begingroup$ Sounds like my job at the moment... $\endgroup$
    – ProfRob
    Jun 23, 2021 at 12:59

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It means exactly what the sentence says. That you were able to do an infinite number of tasks in a finite time. The reason you were able to do it has to do with the nature of the infinite list of tasks. You won't ever be able to do an infinite number of tasks which take one second each to complete.

The nature of the tasks in your book allows them to be completed in a finite time. The tasks are such that their sum of required times converges to a finite number.

Here's anothere infinite list of tasks:

  1. Start at 0cm
  2. Cross the mark at 0.01cm
  3. Cross the mark at 0.161cm
  4. Cross the mark at 0.5cm .

.

.

.

.End at 1cm

The dotted part has all the tasks of the form : "Cross the mark at $x$cm", where $x$ is some rational number between 0 and 1.

You can complete this list of tasks in one second by walking at 1cm per second.

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It just means that the time required was finite (as apposed to infinite). The $1.11111...$ limit may be reached, but it's less than $1.2$, so finite.

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sigma$(n=1, \infty)$ $a_b = \sigma(n=1, \infty)$ $1/2 n^2= 1$ so that means Achilles will eventually complete the task.

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    May 3, 2022 at 5:50

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