sum of infinite series: convergent or divergent??

Question

Sum of infinite series: convergent or divergent?
sum from x=2 to ∞ of 1/(x ln x)

How do I show whether this is convergent or divergent?

I do not need to find the sum if it is convergent. I just need to know how to show convergence or divergence.

Thanks Pascal! I tried to use the integral test earlier but forgot how to integrate 1/(x ln x).

Answer 1

Since this function is positive and strictly decreasing, we can use the integral test -- the series converges iff [2, ∞]∫1/(x ln x) dx converges. However evaluating the integral reveals that [2, ∞]∫1/(x ln x) dx = ln (ln x) |[2, ∞] = ∞, so the series does not converge.

Answer 2

Divergent Sum

Answer 3

ln( n/(2n+5) ) = ln ( 0.5 / (a million+5/(2n) ) = ln(0.5) - ln(a million+2.5/n) as n gets great 2.5n gets small and the latter term gets very small so sum of words from N to M is (M-N)ln(0.5) - O(ln(a million+2.5/N)) ... my O(f) ability some thing of the order of f ... I dont could desire to install writing it precisely through fact i be attentive to that's going to be small so the series can not converge .. even with the shown fact that many words you decide on (N) i will choose an M the place (M-N) is as great as i choose.