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Math Help - does this integral converge?

  1. #1
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    does this integral converge?

    integral( x^p * exp(-x^q)dx, 0,infinite) where, p,q >0
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  2. #2
    Oli
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    Yes it does. Can't remember why though.
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  3. #3
    Super Member PaulRS's Avatar
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    Yep, it converges.

    \int_0^{\infty}x^p\cdot{e^{-x}}dx=\Gamma(p+1)

    To see why it converges, compute the limit \lim_{x\rightarrow{+\infty}}\frac{x^p\cdot{e^{-x^{q}}}}{\frac{1}{x^2}}
    What can you say about \int_1^{\infty}{\frac{dx}{x^2}}? and then what happens with \int_1^{\infty}{x^p\cdot{e^{-x^{q}}}dx} ?
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