volume_mute
The ε-Condition for 1/n
publish date: 2026/05/23 22:07:20.162678 UTC
volume_mute
For any small positive number \(\varepsilon = 0.001\), we want \(1/n < \varepsilon\). What is the minimum value of \(N\) such that \(n > N\) guarantees this?
Correct Answer
N = 1 000
Explanation
We need \(1/n < 0.001\), i.e., \(n > 1/0.001 = 1000\). So choosing \(N = 1000\) works: whenever \(n > 1000\), \(1/n < 0.001 = \varepsilon\). More generally, for any \(\varepsilon > 0\), we choose \(N \ge \lceil 1/\varepsilon \rceil\). This is the precise, quantitative meaning of \(1/n \to 0\) as \(n \to \infty\).
Reference
Introduction to Differential Calculus (Systematic Studies with Engineering Applications for Beginners) - 2012