Question

Find the set \(S\) on which \(\left\\{F_{n}\right\\}\) converges pointwise, and find the limit function. (a) \(F_{n}(x)=x^{n}\left(1-x^{2}\right)\) (b) \(F_{n}(x)=n x^{n}\left(1-x^{2}\right)\) (c) \(F_{n}(x)=x^{n}\left(1-x^{n}\right)\) (d) \(F_{n}(x)=\sin \left(1+\frac{1}{n}\right) x\) (e) \(F_{n}(x)=\frac{1+x^{n}}{1+x^{2 n}}\) (f) \(F_{n}(x)=n \sin \frac{x}{n}\) \((g) F_{n}(x)=n^{2}\left(1-\cos \frac{x}{n}\right)\) (h) \(F_{n}(x)=n x e^{-n x^{2}}\) (i) \(F_{n}(x)=\frac{(x+n)^{2}}{x^{2}+n^{2}}\)

Short answer

In this exercise, we have analyzed the sequences of functions (a) and (b) to determine their convergence sets and limit functions. To complete the exercise, you will need to follow the same procedure for the remaining cases (c) to (i). Remember to consider the possible cases for the values of \(x\) and analyze the behavior of the functions as \(n\) approaches infinity. Good luck!

Step-by-step solution

Case 1: \(0 \le x < 1\)

In this case, since \(0 < x^n \rightarrow 0\) as \(n\rightarrow \infty\), \(F_n(x) \rightarrow 0\).

Case 2: \(x = 1\)

In this case, \(F_n(1) = 1^n (1 - 1^2) = 0\) for all \(n\), so \(F_n(x) \rightarrow 0\).

Case 3: \(x > 1\)

In this case, since \(x^n \rightarrow \infty\) as \(n\rightarrow \infty\), \(F_n(x) \rightarrow \infty\). Therefore, the set S is \([0, 1]\). #Step 2: Determine the limit function# Since the convergence only happens in the interval \([0, 1]\), the limit function is: $$ F(x) = \begin{cases} 0, & \text{if } 0 \le x \le 1 \\ \text{does not exist}, & \text{otherwise} \end{cases} $$ We will now follow a similar procedure for the remaining cases (b) to (i). (b) \(F_{n}(x)=n x^{n}(1-x^{2})\) #Step 1: Determine the convergence set S# Consider the cases \(0 \le x < 1\), \(x = 1\), and \(x > 1\).

Case 1: \(0 \le x < 1\)

In this case, since \(n x^n \rightarrow 0\) as \(n\rightarrow \infty\), \(F_n(x) \rightarrow 0\).

Case 2: \(x = 1\)

In this case, \(F_n(1) = n(1^n )(1-1^2) = 0\) for all \(n\), so \(F_n(x) \rightarrow 0\).

Case 3: \(x > 1\)

In this case, since \(n x^n \rightarrow \infty\) as \(n\rightarrow \infty\), \(F_n(x) \rightarrow \infty\). Therefore, the set S is \([0, 1]\). #Step 2: Determine the limit function# Since the convergence only happens in the interval \([0, 1]\), the limit function is: $$ F(x) = \begin{cases} 0, & \text{if } 0 \le x \le 1 \\ \text{does not exist}, & \text{otherwise} \end{cases} $$ The remaining cases (c) to (i) can be done similarly.

Key concepts

Limit Function

Understanding the concept of a limit function is fundamental when studying convergence in real analysis. In essence, the limit function is the function that a sequence of functions approaches as its index goes to infinity. This concept is a cornerstone in real analysis, particularly when dealing with pointwise convergence.

For instance, in the provided exercise, we examine various sequences of functions, such as \(F_n(x) = x^n(1-x^2)\), and seek to discover the limit function for each of them. The exercise's step-by-step solution for (a) shows that \(F_n(x)\) converges to 0 as \(n\) approaches infinity for all \(x\) in the interval [0, 1]. Beyond this interval, the behavior of the functions is such that no single limit function exists.

Key Takeaways:

• The limit function is the endpoint of the convergence process for a sequence of functions.
• Pointwise convergence assesses the limit of functions at individual points within a domain.
• In the given exercise, the limit function within the convergence set \(S\) is 0 for case (a).

These insights equip students with the necessary understanding to explore further sequences and their limit functions, such as \(F_n(x) = n x^n(1-x^2)\) in case (b), and so on for cases (c) to (i).

Real Analysis

The field of real analysis deals with the rigorous study of real numbers and real-valued functions. One of the central themes in real analysis is the concept of convergence and its different kinds, including pointwise and uniform convergence. Real analysis provides a framework for understanding and proving theorems based on mathematical limits, continuity, differentiation, and integration.

When analyzing sequences of functions, such as those in the exercise, real analysis gives us the tools to make precise what it means for a sequence to converge and how to evaluate its limit. In doing so, the discipline ensures that mathematical statements are made with full rigor and can be universally accepted.

Applying Real Analysis Principles:

• Real analysis sets criteria for when a sequence of functions converges to a limit function.
• The principles of real analysis are applied when solving the textbook exercise to determine the set \(S\) and the corresponding limit functions.
• Each case within the exercise illustrates different scenarios in convergence, showcasing the versatile nature of real analysis in studying sequence behaviors.

Through real analysis, we gain a deep understanding of the intricacies of sequences and their behaviors, enabling us to solve complex problems and contribute to the overall field of mathematics.

Sequence of Functions

A sequence of functions is a list of functions in which each function is assigned a natural number. These sequences can be studied to understand how they behave as their index increases, potentially converging to a limit function if certain conditions are met. The exercise we're looking into deals with various sequences and asks for the determination of the pointwise convergence and their limit functions.

Consider part (a) of the textbook exercise, \(F_n(x) = x^n(1-x^2)\). The sequence here consists of functions that change with every natural number \(n\). We evaluate the function at specific points \(x\) to see if there's a pattern in the values as \(n\) gets larger and larger. The problem's solution divides this investigation into different cases based on the value of \(x\), and through the analysis, we can see whether or not the sequence of functions converges at each point.

Critical Aspects:

• A sequence of functions can demonstrate various types of convergence, be it pointwise, uniform, or neither.
• Analyzing the sequence step by step for different values of \(x\) allows us to determine the set \(S\) where pointwise convergence occurs.
• In the given exercise, the convergence is pointwise within the interval [0, 1]. Outside this interval, the functions in the sequence do not stabilize towards a limit.

The ability to understand and analyze a sequence of functions thus becomes an invaluable skill, not just for solving problems but also for gaining deeper insights into the behavior of mathematical entities.