Question

Find all values of the parameter p for which the integral, 1/(x(lnx)^p) from e to infinity, converges.

Answer 1

let \(u=\ln x\implies du=\frac{dx}x\)
what do you get after integrating?

Answer 2

the integral of 1/(u)^p du?

Answer 3

yes, now integrate...

Answer 4

u^(-p+1)/(-p+1)

Answer 5

right, unless p=1, in which case you get a different case
evaluate what you have for now

Answer 6

think about three possibilities
p>1
p=1
p<1

Answer 7

think about this as\[\lim_{n\to\infty}\left.\frac1{p-1}\frac1{u^{p-1}}\right|_1^n\]

Answer 8

\[=\lim_{n\to\infty}\frac1{p-1}\left(\frac1{n^{p-1}}-1\right)\]again consider three possibilities:
p<1, P=1, p>1

Answer 9

is p>1 because its 1/p-1?

Answer 10

p=1 is a sort of breaking point for the cases for that reason, but also and more importantly because this limit will only converge if the exponent in the denominator is positive, right?

Answer 11

oh I see what you are saying, so yes, you are right
you still have to consider the case when p=1 though

Answer 12

it's pretty trivial though

Answer 13

I mean you should do the other integral though, just to show that it diverges as well

Answer 14

...at least I would to be thorough, but as you prefer

Answer 15

okay thank you!