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) Find the limit superior and limit inferior of each of the following sequences: (a) X = (xn) = (1+(–1)"), n E N (b) X = (xn) = (cos² n E N (c) X = (xn) = 3+ пEN - (rm) = (T) (-1)"n) (d) X = n+1 3.

) Find the limit superior and limit inferior of each of the following sequences: (a) X = (xn) = (1+(–1)"), n E N (b) X = (xn) = (cos² n E N (c) X = (xn) = 3+ пEN - (rm) = (T) (-1)"n) (d) X = n+1 3.

Advanced Engineering Mathematics
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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Basic Real Analysis 1- Answer this

**Exercise 3: Finding Limit Superior and Limit Inferior of Sequences** Determine the limit superior and limit inferior for each of the following sequences: (a) \( X = (x_n) = \left( 1 + (-1)^n \right), n \in \mathbb{N} \) (b) \( X = (x_n) = \left( \cos^2 \frac{n\pi}{2} \right), n \in \mathbb{N} \) (c) \( X = (x_n) = \left( 3 + \frac{(-1)^n}{n} \right), n \in \mathbb{N} \) (d) \( X = (x_n) = \left( \frac{(-1)^n n}{n+1} \right), n \in \mathbb{N} \) **Instructions:** Analyze each sequence to find the limit superior (lim sup) and limit inferior (lim inf).
Transcribed Image Text:**Exercise 3: Finding Limit Superior and Limit Inferior of Sequences** Determine the limit superior and limit inferior for each of the following sequences: (a) \( X = (x_n) = \left( 1 + (-1)^n \right), n \in \mathbb{N} \) (b) \( X = (x_n) = \left( \cos^2 \frac{n\pi}{2} \right), n \in \mathbb{N} \) (c) \( X = (x_n) = \left( 3 + \frac{(-1)^n}{n} \right), n \in \mathbb{N} \) (d) \( X = (x_n) = \left( \frac{(-1)^n n}{n+1} \right), n \in \mathbb{N} \) **Instructions:** Analyze each sequence to find the limit superior (lim sup) and limit inferior (lim inf).
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