Does (lnx)^2/x^1.2 converge or diverge in the limits 1 to infinity

Question

Does the integral converge or diverge? 

 

                      ∫₁^∞(lnx)²/x^1.2 dx

Answer 1

This integral will converge. Generally speaking, we have a ratio of two nondecreasing functions out to infinity.

 

We can make a few quick observations based on the class of the functions involved and their relative rates of growth. For increasing values of x, we will always have:

 

Log(X) << Polynomial(X) << Exp(X)

 

That is, a polynomial will always dominate a logarithm in the long run, similar to how 0.00001x⁵ will eventually overtake 1000000x⁴. The integrand must be forced to zero fast enough that the integral converges.

 

 

We can also just integrate the function and see what we get. Substitute t=ln(x); dt = dx/x; and x=e^t:

 

∫₀^∞ t² e^(-t/5) dt which can be integrated using integration by parts and results in a finite answer.