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Math Help - limit problem...

  1. #1
    Junior Member
    Joined
    Jan 2009
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    india
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    limit problem...

    Q:let fn(x)=n(sinx)^(2n+1)*cos(x)
    then find the value of
    { limt n tends to inf of integral of fn(x) from 0 to pi/2}
    -{integral of limit n tends to inf of fn(x) from 0 to pi/2}
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  2. #2
    Senior Member
    Joined
    Apr 2009
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    Atlanta, GA
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    408

    Fatou's Lemma

    Define f_n(x)=n\sin^{2n+1}(x)\cos(x)

    Find \lim_{n\rightarrow\infty}\int_0^{\pi/2} f_n(x)dx - \int_0^{\pi/2} \lim_{n\rightarrow\infty}f_n(x)dx

    Integrating by simple substitution \int n\sin^{2n+1}(x)\cos(x)dx=\frac{n}{2n+2}\sin^{2n+2}  (x)

    So \lim_{n\rightarrow\infty}\int_0^{\pi/2} f_n(x)dx=\lim_{n\rightarrow\infty}\frac{n}{2n+2}=\  frac12

    But \lim_{n\rightarrow\infty}n\sin^{2n+1}(x)\cos(x)=0, so your answer is zero.
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