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i just ran into this problem and i'm having a hard time solving it:

i would like to know if this integral converges or not, and why.

i'd prefer the normal convergence tests.

$$ \int_2^\infty\frac{x}{\ln^3 x}dx $$

Any help will be greatly appreciated.

thanks,

yaron.

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  • $\begingroup$ $\log x$ grows more slowly than $x$, so you should expect the integrand to fail to go to zero. $\endgroup$ – Gyu Eun Lee May 7 '13 at 4:38
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    $\begingroup$ What are the limits of integration? $\endgroup$ – Mhenni Benghorbal May 7 '13 at 4:41
  • $\begingroup$ 1) the limits of the integration are from 2infinity to infinity. $\endgroup$ – user76508 May 7 '13 at 4:42
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    $\begingroup$ Those limits cannot be right. $\endgroup$ – Pragabhava May 7 '13 at 4:44
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    $\begingroup$ does 2->infinity sound better? $\endgroup$ – user76508 May 7 '13 at 4:48
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Community wiki answer so the question can be marked as answered: No, the integral does not converge, since any power of $\ln x$ grows more slowly than $x$.

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  • $\begingroup$ Would you like to tell why it always goes smaller? (and formal proof?) $\endgroup$ – Charlie Chang Aug 13 '20 at 6:09
  • $\begingroup$ @CharlieChang: If you substitute $x=\mathrm e^y$, it says that $y$ grows more slowly than any power of $\mathrm e^y$, or equivalently (since you can transfer the power to the other side) that any power of $y$ grows more slowly than $\mathrm e^y$. This is a well-known fact about the exponential function. $\endgroup$ – joriki Oct 8 '20 at 9:47

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