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Math Help - Help: Minor detail in proof of uniform convergence

  1. #1
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    Help: Minor detail in proof of uniform convergence

    The problem states:

    Prove that the following limit exists and find the limit.

    lim_{n\rightarrow\infty}\int_0^{\pi/2}1-\frac{\sqrt{x\sin {nx}}}{n}dx

    I've already shown that the pointwise limit is 1, but when I try to show that it converges uniformly using the definition that

    If |f_n(x)-f(x)|<\epsilon

    then it's uniformly convergent, I run into a slight problem... I get to a point where I have

    |\frac{\sqrt x\sqrt{\sin nx}}{n}| and I want to show (after a few more inequalities) that this is less than epsilon.

    However, how can I account for the fact that sometimes the \sqrt{\sin nx} is negative, thus giving me a nonreal value?

    What this all boils down to, is can I say that

    |\frac{\sqrt x\sqrt{\sin nx}}{n}|\leq|\frac{\sqrt x}{n}|
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  2. #2
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    Quote Originally Posted by paupsers View Post
    The problem states:

    Prove that the following limit exists and find the limit.

    lim_{n\rightarrow\infty}\int_0^{\pi/2}1-\frac{\sqrt{x\sin {nx}}}{n}dx

    I've already shown that the pointwise limit is 1, but when I try to show that it converges uniformly using the definition that

    If |f_n(x)-f(x)|<\epsilon

    then it's uniformly convergent, I run into a slight problem... I get to a point where I have

    |\frac{\sqrt x\sqrt{\sin nx}}{n}| and I want to show (after a few more inequalities) that this is less than epsilon.

    However, how can I account for the fact that sometimes the \sqrt{\sin nx} is negative, thus giving me a nonreal value?

    What this all boils down to, is can I say that

    |\frac{\sqrt x\sqrt{\sin nx}}{n}|\leq|\frac{\sqrt x}{n}|
    I am not sure that I follow all of that.
    But \left|\frac{\sqrt x\sqrt{\sin nx}}{n}\right|\leq\left|\frac{\sqrt x}{n}\right| is correct.
    But have you been careful about domain issues? i.e. about \sqrt{\sin nx}
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  3. #3
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    Well, that's another issue. I'm trying to use a theorem that states I can interchange the order of the limit and integral, but only if {f_n} is a sequence of CONTINUOUS functions... Are they not continuous due to this "glitch" with the complex answers?
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