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Math Help - definite integral/ limit of integral

  1. #1
    Junior Member
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    Oct 2009
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    definite integral/ limit of integral

    I need to find the definite integral from 1 to A of f(x)=1/[(x^(1+1/x)] in order to find the limit of this integral as A approaches infinity. I'm not sure if this is analysis or calculus but I just put it here because more than anything I need help finding the integral itself.
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  2. #2
    Super Member
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    I believe it diverges because the function f(x)=\frac{1}{x^{1+\frac{1}{x}}} is asymptotic to g(x)=\frac{1}{x} as x\to\infty since:

    \lim_{x\to\infty} \frac{f(x)}{g(x)}=1

    and therefore the convergence of \int_1^{\infty} f(x)dx is the same as the convergence of \int_1^{\infty}1/x dx

    Here's a nice reference on checking the convergence of improper integrals:

    http://www.math.princeton.edu/~nelson/104/ImproperIntegrals.pdf
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