- #1

- 590

- 0

_{infinity}

[tex]\sum[/tex]1/(ln(n)*n

^{p})

^{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.

[tex]\int[/tex]dx/(ln(x)*x

^{p}) from 2-infnity

i thought of using substitution as

t=ln(x)

x=e

^{t}

dt=dx/x

[tex]\int[/tex]dt/(t*e

^{t(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??