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Math Help - The significant tips for deciding whether improper integral is convergent or not?

  1. #1
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    The significant tips for deciding whether improper integral is convergent or not?

    I will have advanced calculus QUIZ on next week Thursday.

    Matching chapter is

    1. Determine the convergence of improper integral

    2. Technique of integration...

    Mainly, i want to know "intuitively checking Method" , it determines whether improper integral is convergent or not.

    I learn only comparison test. But in solving problem, i can't make decisions well;; for example , i tried to prove given improper integral is divergent even if

    it is convergent....

    Q. SO.. Is there "intuitively checking Method"? If exists, please teach me..!


    ==
    I know other test for deciding..

    thm. Both integral converge or diverge

    thm. dirichlet's test for improper integral...

    But it is just test, it can't help me increase ability of intuitive judgement..
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  2. #2
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    Re: The significant tips for deciding whether improper integral is convergent or not?

    If your function has an anti-derivative that is continuous, then you can use the property that lim f(x) for x approaching a = f(a). If your integral is f(x)dx and it has an anti-derivative F(x), then you can use this known property of limits as applied to continuous functions.

    Basically continuity is defined in analysis by imposing the condition that if you have a function in some open interval (a,b) and the limit of x->c f(x) = f(c) for all values in that interval, then the function is continuous (but the limit from both sides has to give the same value and it has to be finite).

    Now for a limit approaching infinity or negative infinity we only consider a one-sided limit, but this should exist if your anti-derivative is continuous.

    That's the basic intuition of why you can use lim F(a) as a -> infinity if f(x) is a Riemann-Integrable function over some interval.
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