Name: zeno’s paradox and geometric sequences
Uploaded: 2026-06-08T09:38:17.769Z
Duration: 1 min 26 s
Channel: JoVE

设想一个人先走完到门前距离的一半，然后走完剩余距离的一半。同样，再走完剩余距离的一半，此过程持续进行。这种模式产生一系列距离。由于始终存在一个新的中点，这些距离永远不会终止，从而形成一个无限序列。这个思想实验被称为芝诺悖论，它表明如果剩余的距离总是被平分，那么目的地实际上永远无法到达。为了分析此类情况，数学引入了序列的概念——即遵循特定规则的有序数字列表。此处，每个距离称为序列 an 的一项，按确定的顺序排列。在任意步数 n 时的距离等于 1 除以 2 的 n 次幂。随着步骤数的增加，每个距离变得越来越小，但这些项永远不会达到零。由于此过程无限延续，该序列被称为无穷序列。这些序列有助于分析涉及重复、变化和无穷的过程。

序列简介

古希腊哲学家埃利亚的芝诺提出了一系列悖论，以挑战当时关于运动和连续性的主流观念。其中一个悖论设想一个人走向一扇门，但在每一步中仅走过剩余距离的一半。这一系列运动——先走总距离的二分之一，再走四分之一，然后是八分之一，依此类推——构成了一个被称为几何序列的数学概念。每一项均为前一项的一半，可表示为\begin{equation*}a_{n}= \jfrac{1}{2^n}\end{equation*}其中 n 为从 1 开始的步数编号。尽管看起来这个人由于存在无限多个步骤而永远无法到达门口，但数学表明事实并非如此。总距离可以通过对无限序列求和得到。这个从 1/2 开始的几何级数的和为：\begin{equation*}\jfrac{1}{2}+\jfrac{1}{4}+\jfrac{1}{8}+\jfrac{1}{16}+\dots =1\end{equation*}这证实了总距离收敛于1，表明此人最终到达了门口。芝诺悖论凸显了数列与无穷级数在理解极限与连续性方面的有用性。这些数学工具对于分析涉及无限重复或渐进变化的过程至关重要。这一悖论曾是一个哲学难题，如今却展示了数学如何通过精确推理与逻辑分析来解决表面矛盾。

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Education

教育

JoVE Core

Calculus

微积分

Unit

单元

Sequences and Series

数列与级数

KEY TERMS AND CONCEPTS

关键术语与概念

Introduction to Sequences

序列介绍

zeno’s paradox and geometric sequences

Zeno’s paradox shows how geometric sequences can describe motion that keeps getting smaller. In the example, a man walks toward a door but only covers half of the remaining distance each time. That creates a sequence of steps: one-half, one-quarter, one-eighth, and so on.

This pattern is a geometric sequence because each term is half of the previous term. It can be written as a_n = 1/2^n, where n is the step number starting at 1. The terms shrink in a regular way, even though the sequence continues without end.

At first, it may seem that the man never reaches the door because there are infinitely many steps. Mathematics shows a different result. The infinite geometric series 1/2 + 1/4 + 1/8 + 1/16 + ... adds up to 1.

That total means the distance converges to 1, so the man does reach the door. Zeno’s paradox shows how sequences and infinite series help explain limits and continuity. It is a classic example of how mathematics can resolve an apparent contradiction through careful logical reasoning.

Zeno’s Paradox and Geometric Sequences

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Zeno’s paradox shows how geometric sequences can describe motion that keeps getting smaller. In the example, a man walks toward a door but only covers half of the remaining distance each time. That creates a sequence of steps: one-half, one-quarter, one-eighth, and so on. This pattern is a geometric sequence because each term is half of the previous term. It can be written as a_n = 1/2^n, where n is the step number starting at 1. The terms shrink in a regular way, even though the sequence...

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power series and their properties

Power Series as Function Approximations

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interval and radius of convergence

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key terms and concepts

Explore Zeno’s paradox through a geometric sequence, and see how an infinite series can still converge to 1 and explain motion toward a door.

Zeno’s Paradox and Geometric Sequences | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Zeno’s Paradox and Geometric Sequences

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