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Math Help - Comparison Test Proof

  1. #1
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    Comparison Test Proof

    This is my assignment for analysis. When I tried looking more into this theorem, nothing online seems to state this test in a similar way. I don't really understand what the proof is telling me, so I am having a hard time proving it to be true.

    Let g:[a, ∞ ) ->R be a nonnegative function satisfying g is Riemann integrable on [a,c] for every c>a and the integral from a to ∞ g(x) dx <∞ . If f: [a, ∞ ) statisfies
    a) f is Riemann integrable for every c>a and
    b) |f(x)|<= g(x) for all x in [a, ∞ )
    then the improper integral of f on [a, ∞ ) converges and |integral from a to ∞ f(x)|<= the integral from a to ∞ g(x)

    Any help explaining this theorem or starting this proof would be much appreciated.
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  2. #2
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    Re: Comparison Test Proof

    Quote Originally Posted by renolovexoxo View Post
    Let g:[a, ∞ ) ->R be a nonnegative function satisfying g is Riemann integrable on [a,c] for every c>a and the integral from a to ∞ g(x) dx <∞ . If f: [a, ∞ ) statisfies
    a) f is Riemann integrable for every c>a and
    b) |f(x)|<= g(x) for all x in [a, ∞ )
    then the improper integral of f on [a, ∞ ) converges and |integral from a to ∞ f(x)|<= the integral from a to ∞ g(x)

    If f is integrable on [a,c] then |f| is integrable on [a,c] and \left| {\int_a^c {f(x)dx} } \right| \leqslant \int_a^c {\left| {f(x)} \right|dx}  \leqslant \int_a^c {g(x)dx} .
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  3. #3
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    Re: Comparison Test Proof

    How could I start proving that though?
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  4. #4
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    Re: Comparison Test Proof

    Quote Originally Posted by renolovexoxo View Post
    How could I start proving that though?

    Are you saying that your class has not proved the following?

    If f is integrable on [a,c] then |f| is integrable on [a,c] and \left| {\int_a^c {f(x)dx} } \right| \leqslant \int_a^c {\left| {f(x)} \right|dx}.

    If not, then I don't know how to do this problem.
    You must prove that first.
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  5. #5
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    Re: Comparison Test Proof

    We haven't yet, that might why explain why I have no idea how to start. It's due soon though, so any help would be appreciated. I understand how that follows from the triangle inequality.
    Last edited by renolovexoxo; January 28th 2013 at 02:33 PM.
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