# Finding the value of p so that the series converges

find the possible values of p so that the following converges

infinity
$$\sum$$1/(ln(n)*np)
n=2

what i thought of doing was integrating to find the values of p such that the integral will give me an answer not infinity.

$$\int$$dx/(ln(x)*xp) from 2-infnity

i thought of using substitution as
t=ln(x)
x=et
dt=dx/x

$$\int$$dt/(t*et(p-1)) from ln2-infinty

but i have no idea how to continue from here, i think that if p<=1 then my integral will diverge since lim(t->inf) will not be 0, but i am not sure, i know that with a series if lim(n->inf) is not 0 then the series diverges, is this true for integrals as well,
even so, if i know that p<=1 diverges, this does not automatically mean that anything else converges, since lim(n->inf)An=0 doesnt mean necessarily converges, how do i find the values for p where i KNOW the integral diverges??

## Answers and Replies

Your argument works for p <= 1.

To do arguments for p > 1, use a comparison test.. Show that each term (n>2) is less then
1/n^p.

so do i say:
for any n>2 my series is smaller than 1/n^p, and if p>1 1/n^p converge, therefore mine must also diverge, as for p<1 i already proved when i said lim(n->inf) not 0, so my series must diverge, therefore the series only converges when p>1,

do you see a better way of solving this?