• Support PF! Buy your school textbooks, materials and every day products Here!

Limit of an integral

  • Thread starter abcd999
  • Start date
  • #1
4
0

Homework Statement


[tex]lim n \rightarrow\inf \int sin(pi*x^{n})dx[/tex]
...integral is from x=0 to 1/2.


Homework Equations





The Attempt at a Solution


Lebesgue's Dominated Convergence Theorem says that I can move the limit inside, but only if fn converges pointwise to a limit f, which I don't believe it does. Even so, there is no limit as n approaches infinity of fn.
I also tried u substitution, setting u = pi*x^n, but that didn't get me anywhere.

Thanks in advance
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,258
618
Doesn't x^n converge to zero for x in [0,1/2]? Or am I confused?
 
  • #3
4
0
It does, but in order to move the limit inside and use Lebesgue's, doesn't sin(pi*x^n) have to converge to a limit over the entire domain, not just [0,1/2]?
 
  • #4
Dick
Science Advisor
Homework Helper
26,258
618
Not as far as I know. You are only integrating over [0,1/2]. Why do you have to worry about values outside of that range? Just call the domain [0,1/2].
 
  • #5
4
0
I guess I was over thinking it. Thanks for your help.
 

Related Threads for: Limit of an integral

  • Last Post
Replies
4
Views
934
  • Last Post
Replies
2
Views
725
  • Last Post
Replies
8
Views
2K
  • Last Post
Replies
15
Views
2K
Replies
1
Views
617
  • Last Post
Replies
1
Views
3K
Replies
7
Views
3K
Replies
3
Views
2K
Top