Math Problem Statement

Over the given interval

I

determine the sets of uniform convergence for:

f

n

(x)=

x

1+n

x

2

,I=R.

Hint: To find the suprimum of

g

n

(x)=|

f

n

(x)−f(x)|

on

R,

let

t=|x|

for

t≥0.

Another trick, let

u=

n

|x|

and use the inequality

u

1+

u

2

1

2

for all

u∈R.

Question 2

Select one:

a.

f

uniform

0

on

R.

b.

f

uniform

0

on

[−1,1].

c.

f

uniform

0

on

[0,1].

d.

f

uniform

0

on

Solution

We are asked to determine the sets of uniform convergence for the sequence of functions fn(x)=x1+nx2f_n(x) = \frac{x}{1 + nx^2} over the interval I=RI = \mathbb{R}. The goal is to identify the interval on which fn(x)f_n(x) converges uniformly to 0.

Step 1: Pointwise Convergence

First, let's determine the pointwise limit of fn(x)f_n(x) as nn \to \infty.

fn(x)=x1+nx2f_n(x) = \frac{x}{1 + nx^2}

As nn \to \infty, the term nx2nx^2 dominates in the denominator if x0x \neq 0, so:

limnfn(x)=limnx1+nx2=limnxnx2=limn1nx=0\lim_{n \to \infty} f_n(x) = \lim_{n \to \infty} \frac{x}{1 + nx^2} = \lim_{n \to \infty} \frac{x}{nx^2} = \lim_{n \to \infty} \frac{1}{nx} = 0

Thus, fn(x)f_n(x) converges pointwise to 0 for all xRx \in \mathbb{R}.

Step 2: Analyzing Uniform Convergence

To determine whether this convergence is uniform, we analyze the difference fn(x)0=fn(x)|f_n(x) - 0| = |f_n(x)| and find the supremum of this expression over R\mathbb{R}.

We are given a hint to let t=xt = |x| and u=nxu = \sqrt{n}|x|. Rewriting the function in terms of uu, we have:

[ g_n(x) = |f_n(x) - 0| = \left|\frac{x

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Math Problem Analysis

Mathematical Concepts

Uniform Convergence
Pointwise Convergence
Supremum

Formulas

f_n(x) = \frac{x}{1 + nx^2}
g_n(x) = |f_n(x) - 0|

Theorems

-

Suitable Grade Level

Undergraduate Level