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Math Help - does the integral converge for any real n?

  1. #1
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    does the integral converge for any real n?

    Given \int_0^\infty x^n dx find a value for n, which is a real number, such that the limit exists. (Such that the limit converges to some, presumably, real number.)

    I'm omitting the work/algebra I've done to save time/space and because this all seems pretty clear to me.

    It seems pretty obvious, for the case of n\geq1 that it doesn't converge, just by looking at the end behavior of any polynomial function with a positive exponent greater then 1, say 2 and the fact that an integral is the area under a curve. While x^1 is just a constant function and doesn't converge either.

    For 0<n<1, we have some root function, which also doesn't converge. Again I'm just looking at the graph.

    For -1<n<0, this is the inverse of a root. Which looks similar to the inverse of a polynomial. This range appears to converge on the interval (0,1] but diverge on [1,+ \infty] (I'm omitting work here.)

    Since \int \frac{1}{x} dx= ln|x|+c* Yet again I'll just reference the graph and conclude that this diverges over the given interval.

    For n<-1, the function is just an inverse of a polynomial. They converge over the interval [1, + \infty). But all diverge over (0,1].

    So assuming this is as clear as I think it is, a big assumption, and I haven't made any errors . There is no n satisfying the conditions.

    So if i'm correct, and there is no n such that the limit exists/converges, could someone give me some hints as to how I can write this out in a way that isn't so bulky?

    Thanks!
    Last edited by bkbowser; October 23rd 2012 at 05:26 PM. Reason: *i typed out x^-1 strangely.
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  2. #2
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    Re: does the integral converge for any real n?

    actually the problem sheet my teacher handed out specifically mentioned not getting help from the on campus math help center. i really should interpret that to mean professional(ish) help. which means i should extend that restriction to ya'll as well.

    Since I can't delete threads could a moderator help me out with that?
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  3. #3
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    Re: does the integral converge for any real n?

    Is the "on campus math help center" some physical location? If so, it shouldn't be a problem here...

    Hint: There are real values of n such that the integral converges. Maybe you assumed something incorrectly?
    Last edited by richard1234; October 23rd 2012 at 06:44 PM.
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  4. #4
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    Re: does the integral converge for any real n?

    Many universities have a place where struggling calculus students can go to get help with their homework. It's staffed by math graduate students.

    I think that if he's not allowed to go there, then he's correctly interpreted the "spirit" of that rule to mean he's not allowed to post his problem here, either. Admittedly, the "letter" of the rule would mean that he would be allowed to come here.

    - Hollywood
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