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As a picky detail, that's a definite integral.
http://fwd4.me/0lMo

oh you're right I meant improper

ok I've got that. Where is the part where it is improper though?

oh wait nevermind it's the infinity part sorry

then I think I would set t > e and take the integral from e to t

you could do that...then later take the limit as t->infinity

yeah, but how can I take the general integral of that without knowing what p is?

I think I might need to do cases for when p is -1 and when it isn't maybe

apply general power rule
\[\int\limits\limits_{?}^{?} u^{-p} = \frac{u^{-p+1}}{-p+1}\]

ok now all I need to do is see where the lim t-> inf is equal ro inf and when it's not

right...take lim ln(t) and lim 1/ln(t)

I think it's divergent when p<1

if p = then it would also be divergent I think

p=1

yes, 1/0 - 1/0 --> infinity

so as long as the limit doesn't exist or goes to infinity then it's convergent then right?

you mean divergent i think, then yes
if limit exists and is real number, it is convergent

oh right sorry haha backwards again

I understand now, thank you very much for your help!

no problem :)