Question

For what values of p do the integral converge or diverge?

Answer 1

\[\int\limits_{2}^{\infty}\frac{ dx }{ x(lnx)^p }\]

Answer 2

\[\lim_{x \rightarrow \infty }\frac{ 1 }{ (1-p)(lnx)^{p-1} } \]

We see from this that p has to be greater than 1, that way as you approach infinity the limit goes to 0.

p can't equal 1, and I suggest you try several special cases to show to yourself why not. Solve the following easy integrals to get a feel for why.

p=0, p=1, p=-1, p=2, p=1/2

Answer 3

it should me \[\frac{ lnx^{1-p} }{ 1-p }\]

Answer 4

p>1