Generalize improper integral help

[TheKracken]

Homework Statement

Generalize the integral from 0 to 1 of 1/(x^p)
What conditions are necessary on P to make the improper integral converge and not diverge?

I believe I have the answer but I would like to make it more formal and sound. Can someone help me with that?

Homework Equations

None

The Attempt at a Solution

Lim(b--> infinity) of ∫ 1/(x^p)dx (from 1 to b)
= lim(b--> ∞) of ∫ x^(-p)dx (from 1 to b)
= lim(b-->∞) of (x^(-p +1))/(-p+1) evaluated (from 1 to b)
= lim(b--> ∞) of [(b^(-p + 1))/(-p+1) - (1^(-p+1))/(-p+1))]

From here I would normally apply the limit to the last thing listed but it seems there are some constraints immediately needed.

P ≠ 1 also x ≠ 0
how would I figure out what constraints need to be put on P for the improper integral to converge?

By intuition I think I figured out the solution. I would need the [(b^(-p + 1))/(-p+1) portions numerators power to go negative so we can have it tend to 0 rather than infinity.

Therefore I conclude that P>1 for the integral to converge.

 

[Dick]

Homework Statement

Generalize the integral from 0 to 1 of 1/(x^p)
What conditions are necessary on P to make the improper integral converge and not diverge?

I believe I have the answer but I would like to make it more formal and sound. Can someone help me with that?

Homework Equations

None

The Attempt at a Solution

Lim(b--> infinity) of ∫ 1/(x^p)dx (from 1 to b)
= lim(b--> ∞) of ∫ x^(-p)dx (from 1 to b)
= lim(b-->∞) of (x^(-p +1))/(-p+1) evaluated (from 1 to b)
= lim(b--> ∞) of [(b^(-p + 1))/(-p+1) - (1^(-p+1))/(-p+1))]

From here I would normally apply the limit to the last thing listed but it seems there are some constraints immediately needed.

P ≠ 1 also x ≠ 0
how would I figure out what constraints need to be put on P for the improper integral to converge?

By intuition I think I figured out the solution. I would need the [(b^(-p + 1))/(-p+1) portions numerators power to go negative so we can have it tend to 0 rather than infinity.

Therefore I conclude that P>1 for the integral to converge.

The integral of 1/x^2 from 0 to 1 does not converge even though 2>1. I'm really not sure why you are taking the integral from 1 to b and letting b->infinity if you want the integral from 0 to 1.