The chalkboard contains mathematical expressions related to sequences and their limits in the real numbers (denoted as \( \mathbb{R} \)). 1. **Sequences Definitions and Limits:** - \( X = (x_n) \) in \( \mathbb{R}^p \) - \( Y = (y_n) \) in \( \mathbb{R}^p \) These describe sequences \( X \) and \( Y \), with terms \( x_n \) and \( y_n \), respectively, each existing in a p-dimensional real space. 2. **Convergence of Sequences:** - \( x_n \to x \in \mathbb{R} \) - \( y_n \to y \in \mathbb{R} \) This indicates that the sequences \( x_n \) and \( y_n \) converge to the limits \( x \) and \( y \), both of which are real numbers. 3. **Combined Limit Expression:** - \( X \cdot Y = (x_n \cdot y_n) \) - It is stated that \( x_n \cdot y_n \to x \cdot y \) as \( n \) approaches infinity. 4. **Inequality and Result to Show:** - Show \( |x_n y_n - x y| < \epsilon \) The task is to demonstrate that the absolute difference between the product of terms from the sequences and the product of their limits is less than a given small positive value \( \epsilon \). The expressions and tasks are typical for studies on the behavior of sequences and their limits, often encountered in real analysis or advanced calculus courses. There's no graph or diagram; the focus is on the algebraic manipulation of limits and understanding convergence.
The chalkboard contains mathematical expressions related to sequences and their limits in the real numbers (denoted as \( \mathbb{R} \)). 1. **Sequences Definitions and Limits:** - \( X = (x_n) \) in \( \mathbb{R}^p \) - \( Y = (y_n) \) in \( \mathbb{R}^p \) These describe sequences \( X \) and \( Y \), with terms \( x_n \) and \( y_n \), respectively, each existing in a p-dimensional real space. 2. **Convergence of Sequences:** - \( x_n \to x \in \mathbb{R} \) - \( y_n \to y \in \mathbb{R} \) This indicates that the sequences \( x_n \) and \( y_n \) converge to the limits \( x \) and \( y \), both of which are real numbers. 3. **Combined Limit Expression:** - \( X \cdot Y = (x_n \cdot y_n) \) - It is stated that \( x_n \cdot y_n \to x \cdot y \) as \( n \) approaches infinity. 4. **Inequality and Result to Show:** - Show \( |x_n y_n - x y| < \epsilon \) The task is to demonstrate that the absolute difference between the product of terms from the sequences and the product of their limits is less than a given small positive value \( \epsilon \). The expressions and tasks are typical for studies on the behavior of sequences and their limits, often encountered in real analysis or advanced calculus courses. There's no graph or diagram; the focus is on the algebraic manipulation of limits and understanding convergence.
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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Using basic
Transcribed Image Text:The chalkboard contains mathematical expressions related to sequences and their limits in the real numbers (denoted as \( \mathbb{R} \)).
1. **Sequences Definitions and Limits:**
- \( X = (x_n) \) in \( \mathbb{R}^p \)
- \( Y = (y_n) \) in \( \mathbb{R}^p \)
These describe sequences \( X \) and \( Y \), with terms \( x_n \) and \( y_n \), respectively, each existing in a p-dimensional real space.
2. **Convergence of Sequences:**
- \( x_n \to x \in \mathbb{R} \)
- \( y_n \to y \in \mathbb{R} \)
This indicates that the sequences \( x_n \) and \( y_n \) converge to the limits \( x \) and \( y \), both of which are real numbers.
3. **Combined Limit Expression:**
- \( X \cdot Y = (x_n \cdot y_n) \)
- It is stated that \( x_n \cdot y_n \to x \cdot y \) as \( n \) approaches infinity.
4. **Inequality and Result to Show:**
- Show \( |x_n y_n - x y| < \epsilon \)
The task is to demonstrate that the absolute difference between the product of terms from the sequences and the product of their limits is less than a given small positive value \( \epsilon \).
The expressions and tasks are typical for studies on the behavior of sequences and their limits, often encountered in real analysis or advanced calculus courses. There's no graph or diagram; the focus is on the algebraic manipulation of limits and understanding convergence.
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