**Sequence and Limit Point Analysis**Consider the sequence defined by $ x_n = 1 - \frac{1}{n} $.- **Graphical Representation:** This sequence can be visualized on the real number line. As $ n $ increases, the value of $ x_n $ approaches 1. You would observe that for each successive term, the value gets closer to 1 but never actually equals 1.- **Open Set around the Limit Point:** The limit point for this sequence is 1. To define an open set $ U $ around this limit point, we can choose an open interval (for example, $ U = (0.9, 1.1) $). For $ N = 3 $, all $ x_n $ for $ n \geq N $ are within this open interval $ U $. This implies that starting from $ n = 3 $ onwards, the terms are inside the interval $ (0.9, 1.1) $, indicating that the sequence eventually stays within any neighborhood around the limit point as $ n $ progresses.This task involves understanding how sequences behave as they progress and how to describe their convergence using limit points and neighborhoods in the context of real analysis.

Name: **Sequence and Limit Point Analysis**Consider the sequence defined by
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Description: **Sequence and Limit Point Analysis**Consider the sequence defined by $ x_n = 1 - \frac{1}{n} $.- **Graphical Representation:** This sequence can be visualized on the real number line. As $ n $ increases, the value of $ x_n $ approaches 1. Y

Given sequence is xn=1-1n Then, x1=1-1=0, x2=1-12=12, x3=1-13=23, x4=1-14=34 and so on. That means…

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Consider the sequence dened by Xn = 1 – 1/n. Make a sketch of this sequence on the real number line. Provide an open set U (an open interval for example) around the limit point of this sequence such that for N = 3 all Xn are in U for all n 2 N.

Consider the sequence dened by Xn = 1 – 1/n. Make a sketch of this sequence on the real number line. Provide an open set U (an open interval for example) around the limit point of this sequence such that for N = 3 all Xn are in U for all n 2 N.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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**Sequence and Limit Point Analysis**

Consider the sequence defined by $ x_n = 1 - \frac{1}{n} $.

- **Graphical Representation:**
This sequence can be visualized on the real number line. As $ n $ increases, the value of $ x_n $ approaches 1. You would observe that for each successive term, the value gets closer to 1 but never actually equals 1.

- **Open Set around the Limit Point:**
The limit point for this sequence is 1. To define an open set $ U $ around this limit point, we can choose an open interval (for example, $ U = (0.9, 1.1) $).

For $ N = 3 $, all $ x_n $ for $ n \geq N $ are within this open interval $ U $. This implies that starting from $ n = 3 $ onwards, the terms are inside the interval $ (0.9, 1.1) $, indicating that the sequence eventually stays within any neighborhood around the limit point as $ n $ progresses.

This task involves understanding how sequences behave as they progress and how to describe their convergence using limit points and neighborhoods in the context of real analysis.

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