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Math Help - Proving this integral converges

  1. #1
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    Proving this integral converges

    SEE EDIT BELOW.

    Show that \int_0^\infty e^{-x^2} converges.

    e^{-x^2} < e^{-x} for x>1.

    \int_0^\infty e^{-x} converges to 1.

    So, by the Comparison Theorem, \int_0^\infty e^{-x^2} converges.

    Does that work? I'm only wary of it because I used the fact that e^{-x^2} < e^{-x} for x>1 which doesn't consider x between 0 and 1... Any help?

    EDIT: Sorry, I misread the problem! I'm supposed to do \frac{1}{2}\int_0^\infty e^{-x^2} which fixes my issue.
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  2. #2
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    Krizalid's Avatar
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    well, we simply split this thing on two sets: [0,1] and [1,\infty), and we can proceed from there.
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