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Help needed with an integration and limits problem! watch

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    Hi there guys,
    I'm stuck on this problem at uni involving limits and integrals. Any help would be greatly appreciated, and a step by step solution would be even better.

    Many thanks in advance,
    Kitrak

    BTW, the question is attached as an image


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    integrate x^(-a) as you would normally. then think about for which values of a the integrated function makes sense.

    Hint:
    Spoiler:
    Show
    a>1
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    i don't understand, sorry.
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    I mean, once i integrate, what do i do from there?
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    any help?
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    This seems a very interesting question, would love to see an answer to this.
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    any help guys?!
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    When a = 2 we have,
    

\displaystyle \\

I = \int^\infty_1 x^{-2} dx  \\

I = \lim_{N \to \infty} \int_1^N x^{-2} dx \\

I = \lim_{N \to \infty} \left[-x^{-1}\right]_1^N \\

I = \lim_{N \to \infty} -\frac{1}{N} + 1 \\

I = 1
    so the integral converges in this case
    Try it for other values of a (say 0.5, 1) at see what happens
    Then try a more general proof
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    what type of general proof do i then do?
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    any ideas guys?
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    Can you find \int_{1}^{N}  x^{-a} dx?
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    Remember, you have your bog standard rule:

    \int x^n dx = \frac{x^{n + 1}}{n + 1} (+ c) (or, in words, add one to the power, then divide by the power.)

    Now, can you apply this rule to \int x^{-a} dx?

    Once you've found this, plug in the limits of integration, N and 1.

    Finally, find the values of a for which your answer doesn't -> infinity when N -> infinity.
 
 
 
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Updated: November 17, 2008

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