(c) (#n) = (n cos“ ()), n e N. %3D 2 (d) (xn) = ((-1)" - n+1 n E N. %3D

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Find the limit superior and limit inferior of each of the following sequences:

The image depicts a set of mathematical sequences, denoted as \((x_n)\), each with different formulas defined for \(n \in \mathbb{N}\) (natural numbers).

**(b)** \((x_n) = \left(n \cos\left(\frac{n\pi}{2}\right)\right), n \in \mathbb{N}.\)

In this sequence, each term is calculated by multiplying \(n\) by the cosine of \(\frac{n\pi}{2}\).

**(c)** \((x_n) = \left(n \cos^2\left(\frac{n\pi}{2}\right)\right), n \in \mathbb{N}.\)

Here, each term is \(n\) times the square of the cosine of \(\frac{n\pi}{2}\).

**(d)** \((x_n) = \left((-1)^n - \frac{n}{n+1}\right), n \in \mathbb{N}.\)

This sequence alternates in sign based on the expression \((-1)^n\) and subtracts a fraction \(\frac{n}{n+1}\) from it.

These sequences explore the interplay between trigonometric functions and fractional expressions.
Transcribed Image Text:The image depicts a set of mathematical sequences, denoted as \((x_n)\), each with different formulas defined for \(n \in \mathbb{N}\) (natural numbers). **(b)** \((x_n) = \left(n \cos\left(\frac{n\pi}{2}\right)\right), n \in \mathbb{N}.\) In this sequence, each term is calculated by multiplying \(n\) by the cosine of \(\frac{n\pi}{2}\). **(c)** \((x_n) = \left(n \cos^2\left(\frac{n\pi}{2}\right)\right), n \in \mathbb{N}.\) Here, each term is \(n\) times the square of the cosine of \(\frac{n\pi}{2}\). **(d)** \((x_n) = \left((-1)^n - \frac{n}{n+1}\right), n \in \mathbb{N}.\) This sequence alternates in sign based on the expression \((-1)^n\) and subtracts a fraction \(\frac{n}{n+1}\) from it. These sequences explore the interplay between trigonometric functions and fractional expressions.
Expert Solution
Step 1

To find the limit superior and limit inferior of each of the following sequences:

xn=ncos2, nN

xn=ncos22, nN

xn=(1)nnn+1, nN

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