MHB Does This Sequence Converge Uniformly?

  • Thread starter Thread starter mathmari
  • Start date Start date
  • Tags Tags
    Convergence
mathmari
Gold Member
MHB
Messages
4,984
Reaction score
7
Hey! :giggle:

We have the sequence of functions $$f_n=\sin (x)-\frac{nx}{1+n^2}$$ I want to check the pointwise andthe uniform convergence.

We have that $$f^{\star}(x)=\lim_{n\rightarrow \infty}f_n(x)=\lim_{n\rightarrow \infty}\left (\sin (x)-\frac{nx}{1+n^2}\right )=\sin(x)$$ So $f_n(x)$ converges pointwise to$f^{\star}=\sin(x)$.
We have that $$\left |f_n(x)-f^{\star}(x)\right |=\left |\sin (x)-\frac{nx}{1+n^2}-\sin(x)\right |=\left |-\frac{nx}{1+n^2}\right |$$ We have to calculate first the supremum for $x\in \mathbb{R}$ and then the limit for $n\rightarrow \infty$.
Isn't the supremum $x\in \mathbb{R}$ the infinity? :unsure:
 
Physics news on Phys.org
mathmari said:
Isn't the supremum $x\in \mathbb{R}$ the infinity?
Hey mathmari!

Yes, it is. (Nod)
 
Klaas van Aarsen said:
Yes, it is. (Nod)

So $f_n$ doesn't converge uniformly to$f^{\star}$, right? :unsure:
 
mathmari said:
So $f_n$ doesn't converge uniformly to$f^{\star}$, right?

Indeed. :geek:
 
Comment on Grammer: "uniformly" is an adverb and so modifies to verbs, adjectives, and other adverbs. Here "converge" is a noun and so requires the adjective "uniform".

One can ask "Does this converge uniformly?" or "Is this convergence uniform?" but not "Is this convergence uniformly".

(Yes, I realize this was probably just a typo but I couldn't help myself!)
 

Similar threads

Replies
9
Views
2K
Replies
21
Views
2K
Replies
3
Views
2K
Replies
1
Views
1K
Replies
17
Views
971
Replies
6
Views
2K
Replies
2
Views
2K
Back
Top