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Math Help - Limit of power series of discontinuous function

  1. #1
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    Limit of power series of discontinuous function

    Fn(x)=|Sin^n(X)| is convergent to F(x) where F(x) =1 when x=(m+1/2)pi and F(x) =0when x =otherwise

    prove \lim n\rightarrow \inf   \int |F(x)-Fn(x)|dx  \ne 0 integrate from -infinity to +ve infinity


    1. not sure how to integrate a function that is discontinuous at a single point
    2 my main confusion is Lim n->inf Fn(x) = 0 which is equal to F(x). Intuitively I can tell the area under sin^n X is not zero even when n->infinity. but I am not sure how to prove it using mathematical formulae
    Last edited by parklover; August 27th 2011 at 12:14 PM.
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  2. #2
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    Re: Limit of power series of discontinuous function

    What have you tried?

    Where are you stuck?
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  3. #3
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    Re: Limit of power series of discontinuous function

    Quote Originally Posted by parklover View Post
    Fn(x)=|Sin^n(X)| is convergent to F(x) where F(x) =1 when x=(m+1/2)pi and F(x) =0when x =otherwise

    prove \lim n\rightarrow \inf   \int |F(x)-Fn(x)|dx   !=0 integrate from -infinity to +ve infinity


    1. not sure how to integrate a function that is discontinuous at a single point
    2 my main confusion is Lim n->inf Fn(x) = 0 which is equal to F(x). Intuitively I can tell the area under sin^n X is not zero even when n->infinity. but I am not sure how to prove it using mathematical formulae
    F(x) \ne 0 on a set of measure zero, so as far as integration is concerned you can replace it by zero.

    CB
    Last edited by CaptainBlack; August 27th 2011 at 02:05 PM.
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  4. #4
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    Re: Limit of power series of discontinuous function

    Quote Originally Posted by CaptainBlack View Post
    F_n(x) \ne 0 on a set of measure zero, so as far as integration is concerned you can replace it by zero.

    CB
    sorry ,not really understand. you mean set F_n(x) = 0 when n->infinity and integrate? How can you get end result not equal 0?
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  5. #5
    Grand Panjandrum
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    Re: Limit of power series of discontinuous function

    Quote Originally Posted by parklover View Post
    sorry ,not really understand. you mean set F_n(x) = 0 when n->infinity and integrate? How can you get end result not equal 0?
    Sorry, typo I mean that F(x)\ne 0 on a set of measure zero, so:

    \lim_{n \to \infty}\int_{-\infty}^{\infty}|F(x)-F_n(x)|dx=\lim_{n \to \infty}\int_{-\infty}^{\infty}|F_n(x)|dx

    CB
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