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Math Help - Series convergence

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    Series convergence

    Hello I have the series k*e^(-k^2) from one to infinity. By the integral test I find that the series diverges since the integral diverges to minus infinity (did the integral by substitution). By ratio test I reached that the a(k+1)/a(k) = (1+1/k) * e^(-2k-1) which tends to 0<1 as k tends to infinity so it converges by ratio test. What's happening here? I checked that I can use the integral test since my function is continuous, positive and non-increasing. With the integral test i reached the point -1/2 * [e^u] (with -1 as lower limit after the integration and + infinity as the upper limit after the integration) And finally I had -1/2 * infinity = - infinity so the improper integral doesn't exist since it is not finite and so our series diverges. Can anyone help me finding what am I doing wrong? Thanks in advance!!
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    Quote Originally Posted by Darkprince View Post
    Hello I have the series k*e^(-k^2) from one to infinity. By the integral test I find that the series diverges since the integral diverges to minus infinity.
    Rethink that intgral.
    \int {xe^{ - x^2 } dx}  = \frac{{ - 1}}{2}e^{ - x^2 }.
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    Quote Originally Posted by Darkprince View Post
    Hello I have the series k*e^(-k^2) from one to infinity. By the integral test I find that the series diverges since the integral diverges to minus infinity (did the integral by substitution).

    The integral converges

    \int_1^{+\infty}xe^{-x^2}\;dx=\left[-e^{-x^2}/2\right]_1^{+\infty}=1/2e


    Edited: Sorry, I didn't see Plato's post.
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    I set -x^2 = u, then -2xdx=du so xdx = -1/2 * du. limit 1 becomes -1 so we have to integrate from -1 to infinity of (-1/2)*e^u * du. This becomes (-1/2)*[e^infinity - e^(-1)] which equals to minus infinity. Am I wrong to that? I am trying to observe what went wrong with my calculation.
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    Quote Originally Posted by Darkprince View Post
    I set -x^2 = u, then -2xdx=du so xdx = -1/2 * du. limit 1 becomes -1 so we have to integrate from -1 to infinity of (-1/2)*e^u * du. This becomes (-1/2)*[e^infinity - e^(-1)] which equals to minus infinity. Am I wrong to that? I am trying to observe what went wrong with my calculation.
    You forgot to change the limits of integration. If x goes from 1 to ∞, then u goes from –1 to –∞.
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    Oh yes, you are right! It was an exam exercise I did today! I only changed the one limit of integration!!! I did from -1 to infinity instead of -1 to - infinity. So probably all marks will be lost?
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    Quote Originally Posted by Darkprince View Post
    So probably all marks will be lost?
    That would be a bit harsh. You used a good method and only made one error (though it led to the wrong answer). I would expect that you would get some partial credit from most examiners.
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    Thank you very much Opalg! Silly me! I should have done the ratio test from the beginning, it's easier and with less possibilities for mistakes, also I wouldn't have to state conditions to use it. With the integral test I had to state the three conditions, do integration by substitution and at the top reach the wrong result Anyways, thank you all for the answers and by the way I don't know why I received an infraction. This type of question we have done it all year in ACF (Analytical and Computational Foundations), not Calculus. Most of my infractions resulted from this type of mistakes I did!
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