Integral Test: Convergence/Divergence of $\sum_{x=2}^\infty (\ln x)^{-1}$

[bp_psy]

Homework Statement

Determine convergence or divergence using the integral test for.
$$\sum _{x=2}^{\infty } \left( \ln \left( x \right) \right) ^{-1}$$

Homework Equations

i should take the limit as b goes to infinity of

$$\int _{2}^{b}\! \left( \ln \left( x \right) \right) ^{-1}{dx}$$

The problem is that the function is not integrable.I know there was trick show that it is divergent but I don't remember what it was.

The Attempt at a Solution

 

[LCKurtz]

When you say it isn't integrable, do you mean the integral diverges or that you can't find an antiderivative?

The integral and the series converge or diverge togther.

 

[bp_psy]

When you say it isn't integrable, do you mean the integral diverges or that you can't find an antiderivative?

The integral and the series converge or diverge togther.

I can't find the antiderivative. Maple give's me $$-{\it Ei} \left( 1,-\ln \left( x \right) \right)$$ and i don't know what it means. I am currently doing cal 3 but I think remember this exercise from cal 2. Is there any other test that I could use to show the divergence?

 

[Bohrok]

I would do a comparison with 1/√x.

 

[aPhilosopher]

ln(2ⁿ) = n*ln(2) < n so 1/ln(2ⁿ) > 1/n if that helps.

 

[bp_psy]

I would do a comparison with 1/√x.

Thanks. It is divergent by the comparison test.