# integral convergence - does this integration converge

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.

• $\log x$ grows more slowly than $x$, so you should expect the integrand to fail to go to zero. – Gyu Eun Lee May 7 '13 at 4:38
• What are the limits of integration? – Mhenni Benghorbal May 7 '13 at 4:41
• 1) the limits of the integration are from 2infinity to infinity. – user76508 May 7 '13 at 4:42
• Those limits cannot be right. – Pragabhava May 7 '13 at 4:44
• does 2->infinity sound better? – user76508 May 7 '13 at 4:48

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 $x$.
• @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. – joriki Oct 8 '20 at 9:47