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Thread: Improper integral

  1. #1
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    Improper integral

    Hi,
    Determen whether the integral of(x-[x]-0.5)/ln(x) from 2 to infinity converges,absolutely converges or diverges,
    where [x] denotes the lower integer part of x
    Thank's in advance

    I tried to write the integral as the infinite sum of integrals from n to n+1 of (x-n-0.5)/ln(x) but so far with no specific consequences.
    also, the numerator in each interval [n,n+1] is symmetric around x=n+0.5 so translation may help.
    I need some help
    Thank's again











    HI,
    Determen whether the integral of(x-[x]-0.5)/ln(x) from 2 to infinity converges,absolutely converges or diverges,
    where [x] denotes the lower integer part of x
    Thank's in advance

    I tried to write the integral as the infinite sum of integrals from n to n+1 of (x-n-0.5)/ln(x) but so far with no specific consequences.
    also, the numerator in each interval [n,n+1] is symmetric around x=n+0.5 so translation may help.
    I need some help
    Thank's again
    Last edited by hedi; Apr 19th 2019 at 06:03 PM.
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  2. #2
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    Re: Improper integral

    For the sum of integrals I got $\displaystyle \sum_{n=2}^\infty\int_n^{n+1}\left(\frac{x}{\ln x} - \frac{n}{\ln x} - \frac{1}{2\ln x}\right) dx$.

    The denominator breaks the symmetry such that the absolute value at the upper bound will be smaller than that at the lower bound.

    Spoilers: https://www.wolframalpha.com/input/?...rate+x%2F(lnx) , https://www.wolframalpha.com/input/?...rate+1%2F(lnx)

    Is the original function even integrable in the range $\displaystyle [2, \infty)$? It has an infinite number of discontinuities in that range (a function bounded on a closed interval and with a finite number of discontinuities on that interval is integrable). I suspect breaking it up as that sum is valid, but I'm not a mathematician.
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