Convergence of an Integral with Increasing Exponent

[abcd999]

Homework Statement

lim n \rightarrow\inf \int sin(pi*x^{n})dx
...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

 

[Dick]

Doesn't x^n converge to zero for x in [0,1/2]? Or am I confused?

 

[abcd999]

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]?

 

[Dick]

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].

 

[abcd999]

I guess I was over thinking it. Thanks for your help.