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Math Help - trouble with evaluating convergence of an integral

  1. #1
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    trouble with evaluating convergence of an integral

    for what values of the real number x is the improper integral [integral from 1 to infinity] t^(x-1)e^(-t)dt convergent?

    my solution was:


    1<= t < infinity

    suppose x < 1

    and define f(t) = t^(x-1)e^(-t) = e^(-t)/t^(1-x) , g(t) = e^(-t)
    now x < 1, so x-1 < 0, which implies that 1 - x > 0

    and since we are on the interval 1 <= t < infinity, 1/t^(1-x) <= 1
    so f(t) <= g(t) for t >= 1, and by the comparison theorem if g(t) converges then so does f(t)

    [integral from 1 to infinity] g(t) = [integral from 1 to infinity] e^(-t)dt = 1/e

    so g(t), converges for all x < 1, and by the comparison theorem so does f(t)

    I used a similar proof using L'hospital's rule t prove the case for x >= 1
    but my question is: is this right (for x < 1)??


    because say I had chosen g(t) = 1/t^(1-x)

    g(t) >= f(t) for all t >= 1 because f(t) = e^(-t)/t^(1-x) and e^(-t) = 1/e^t < 1 for t >=1

    so using the comparison theorem with this g(t), if g(t) converges then so does f(t)

    [integral from 1 to infinity] g(t) = [integral from 1 to infinity] 1/t^(1-x) *dt

    which converges if 1-x > 1 ==> x < 0,
    and diverges if 1-x <= 1 ==> 0 <= x

    so I'm getting two different answers!! what am I doing wrong!?!?
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  2. #2
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    Quote Originally Posted by minivan15 View Post
    for what values of the real number x is the improper integral [integral from 1 to infinity] t^(x-1)e^(-t)dt convergent?
    Notice that t^{x-1} \leq e^{t/2} for t\geq T for some T\geq 1 because exponentials overtake powers.
    If makes no difference what x is.

    Therefore, \int_1^{\infty} t^{x-1} e^{-t} dt = \int_1^T t^{x-1}e^{-t}dt + \int_T^{\infty}t^{x-1}e^{-t/2}dt (if and only if the integral converges).

    Now,
    \int_T^{\infty} t^{x-1} e^{-t} dt \leq \int_T^{\infty} e^{-t/2} dt < \infty
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  3. #3
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    okay, I understand this!

    but can you tell why my method contradicts itself? I'm okay with your answer I'd just like to see where I went wrong with my method.
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