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Math Help - Adv Calc integral

  1. #1
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    Adv Calc integral

    Need some help on this.

    Improper Reimann Integration


    Suppose that f: [1,) R is locally integrable
    on [1,) and f(x)0 for all x in [1,). Prove that
    ∫(1 to ∞)f(x)dx converges iff there existsM > 0 s.t.
    ∫(from 1 to x) f(t)dt M for all x in [1,).

    f is said to be locally integrable on (a,b) iff f is integrable on each closed subinterval [c,d] of (a,b).

    Thanks.

    Last edited by taypez; November 26th 2007 at 09:48 AM. Reason: Clarity :-)
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  2. #2
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    Please review your post. I think that there may be a typo in the definition of the function. Also, what is the definition of ‘locally integrable’?
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  3. #3
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    Need some help on this.

    Improper Reimann Integration


    Suppose that f: [1,) R is locally integrable
    on [1,) and f(x)0 for all x in [1,). Prove that
    ∫(1 to ∞)f(x)dx converges iff there existsM > 0 s.t.
    ∫(from 1 to x) f(t)dt M for all x in [1,).

    f is said to be locally integrable on (a,b) iff f is integrable on each closed subinterval [c,d] of (a,b).

    Thanks.
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  4. #4
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    It is hard to know what there is to prove here.
    Note that because f(t) > 0 you know that F(x) = \int\limits_1^x {f(t)dt} is an increasing function. Knowing that the integral converges let M be the limit. Therefore, M is the bound. If we know that the integral is bounded by some M then use sequences to show that the integral converges. (Bounded increasing sequences converge.)
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