Find the limit superior and limit inferior of each of the following sequences:
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.
Branch of mathematical analysis that studies real numbers, sequences, and series of real numbers and real functions. The concepts of real analysis underpin calculus and its application to it. It also includes limits, convergence, continuity, and measure theory.
Expert Solution
Step 1
To find the limit superior and limit inferior of each of the following sequences:
