follow all instructions, thanks iii) lim sup (n)Observe that {n} is a bounded sequence.• Explain why= 1.Conclude thatlim sup (n) = sup Slim sup (n) = 1.

According to the question The notationrepresents the limit superior of a…

Answered: iii) lim sup (x₂) = 1. Observe that {n}…

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Step 1: The limit superior is the largest limit point

According to the question The notation $space " l i m space s u p space left parenthesis n right parenthesis " space$ represents the limit superior of a sequence $n.$ The limit superior is the largest limit point or accumulation point of the sequence. In your case, it's given that $l i m space s u p space left parenthesis n right parenthesis space equals space 1.$
1. Bounded Sequence: we observed that $left curly bracket n right curly bracket$ is a bounded sequence. This means that there exists a real number M such that for all n in the sequence, we have $space vertical line n vertical line space less or equal than space M. space$ In other words, the values in the sequence $left curly bracket n right curly bracket$ do not grow arbitrarily large; they are bounded above by $M.$
2. Definition of Limit Superior: The limit superior of a sequence $left curly bracket n right curly bracket$ is the supremum (sup) of the set S, where S consists of all the limit points of the sequence. A limit point is a real number L such that for any $epsilon space greater than 0$ , there exists an N such that for all $space n space greater or equal than space N comma space vertical line n space minus space L vertical line space less than space epsilon. space$ In simpler terms, it's a point that the sequence gets arbitrarily close to as n becomes large.