Determine the following: (a) (b) 8 () and (9) n -1 n+1 U["¹"+¹] and ["¹" n n no] n+1 n
Determine the following: (a) (b) 8 () and (9) n -1 n+1 U["¹"+¹] and ["¹" n n no] n+1 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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![### Problem 1.46
**Determine the following:**
#### (a)
\[
\bigcup_{n=1}^{\infty} \left( \frac{1}{n} - \frac{1}{n} \right) \quad \text{and} \quad \bigcap_{n=1}^{\infty} \left( \frac{1}{n} - \frac{1}{n} \right)
\]
#### (b)
\[
\bigcup_{n=1}^{\infty} \left[ \frac{n-1}{n}, \frac{n+1}{n} \right] \quad \text{and} \quad \bigcap_{n=1}^{\infty} \left[ \frac{n-1}{n}, \frac{n+1}{n} \right]
\]
---
For problem (a), the expressions involve the union and intersection of a sequence of sets where each set is of the form \(\left( \frac{1}{n} - \frac{1}{n} \right)\). Evaluating the terms inside the parentheses will provide insight into how to proceed with the union and intersection operations.
For problem (b), each set is an interval of the form \(\left[ \frac{n-1}{n}, \frac{n+1}{n} \right]\). The goal is to determine the union and intersection of these intervals as \(n\) approaches infinity.
---
**Note**: When solving these problems, recall the properties of sequences, intervals, and set operations like unions and intersections. The behavior of \( \frac{1}{n} \) as \( n \) approaches infinity will be particularly insightful for determining the limits of these operations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f7f7e91-311f-40b2-8ef2-058480eb16a0%2Fb41eaeae-268c-4f00-be5c-a06a001b8bc2%2F7zb8ty_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 1.46
**Determine the following:**
#### (a)
\[
\bigcup_{n=1}^{\infty} \left( \frac{1}{n} - \frac{1}{n} \right) \quad \text{and} \quad \bigcap_{n=1}^{\infty} \left( \frac{1}{n} - \frac{1}{n} \right)
\]
#### (b)
\[
\bigcup_{n=1}^{\infty} \left[ \frac{n-1}{n}, \frac{n+1}{n} \right] \quad \text{and} \quad \bigcap_{n=1}^{\infty} \left[ \frac{n-1}{n}, \frac{n+1}{n} \right]
\]
---
For problem (a), the expressions involve the union and intersection of a sequence of sets where each set is of the form \(\left( \frac{1}{n} - \frac{1}{n} \right)\). Evaluating the terms inside the parentheses will provide insight into how to proceed with the union and intersection operations.
For problem (b), each set is an interval of the form \(\left[ \frac{n-1}{n}, \frac{n+1}{n} \right]\). The goal is to determine the union and intersection of these intervals as \(n\) approaches infinity.
---
**Note**: When solving these problems, recall the properties of sequences, intervals, and set operations like unions and intersections. The behavior of \( \frac{1}{n} \) as \( n \) approaches infinity will be particularly insightful for determining the limits of these operations.
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