Uniform Convergence of nx/nx+1

In summary, the function f_n(x)=\frac{nx}{nx+1} converges pointwise to 0 at x=0 and 1 at all other points on [0,\infty). It is not uniformly convergent on [0,1] as Limit_{n\rightarrow\infty} sup |f_n(x)-1| is not 0. However, it does converge uniformly on the interval (0,1] and to show this, we need to prove that Limit_{n\rightarrow\infty} sup |f_n(x)-1|=0. The issue with the latex formatting may be due to it being in all capitals.
  • #1
hmmmmm
28
0
I am given [tex]f_n(x)=\frac{nx}{nx+1}[/tex] defined on [tex] [0,\infty) [/tex] and I have that the function converges pointwise to [tex] 0 \ \mbox{if x=0 and} 1\ \mbox{otherwise}[/tex]

Is the function uniform convergent on [tex] [0,1] [/tex]?

No. If we take x=1/n then [tex]Limit_{n\rightarrow\infty}|\frac{1/n*n}{1+1/n*n}-1|=0.5[/tex]

which implies that [tex]Limit_{n\rightarrow\infty} sup |f_n(x)-1|[/tex] is not 0.

I am then asked if it converges uniformly on the interval [tex](0,1][/tex] which I think it does but how do I show that [tex]Limit_{n\rightarrow\infty} sup |f_n(x)-1|[/tex]=0?

thanks for any help
 
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  • #2
Your latex is screwed up.
 
  • #3
Yeah do you know why that is?

thanks for any help
 
  • #4
hmmmmm said:
Yeah do you know why that is?

thanks for any help

don't put TEX in capitals.
 

Related to Uniform Convergence of nx/nx+1

What is the definition of uniform convergence?

Uniform convergence is a type of convergence in which the rate of convergence is independent of the point in the domain. In other words, as the independent variable approaches a certain value, the function approaches the same value at the same rate, regardless of the point in the domain.

How is uniform convergence different from pointwise convergence?

Pointwise convergence is a type of convergence in which the rate of convergence can vary depending on the point in the domain. In contrast, uniform convergence has a constant rate of convergence, regardless of the point in the domain.

What is the significance of uniform convergence in mathematical analysis?

Uniform convergence is important in mathematical analysis because it allows for the manipulation of limits and integrals, making it easier to evaluate complex functions. It also allows for the use of stronger theorems and techniques in proving the convergence of a series or function.

Can a function be uniformly convergent but not pointwise convergent?

Yes, a function can be uniformly convergent but not pointwise convergent. This is because pointwise convergence only requires that the function approaches the same value at each point in the domain, while uniform convergence requires a constant rate of convergence at each point.

How is the uniform convergence of nx/nx+1 related to the uniform convergence of 1/n?

The uniform convergence of nx/nx+1 is related to the uniform convergence of 1/n in that they both have a constant rate of convergence. However, the function nx/nx+1 has a larger rate of convergence compared to 1/n, making it a stronger condition for uniform convergence.

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