Series. Precise definition of the Direct Comparison Test? $$\sum_{n=2}^\infty \frac{\ln n}{n} $$
I see direct comparison:
$$ \frac{\ln(n)}{n} > \frac{1}{n} \qquad (\text{for } n>2) $$
But, this is not true, since $\ln x >1$ only when $x>e = 2.71$
So, for $n = 2.5$, it's not greater.
So, what is the exact definition of the direct comparison test?  The inequality only needs to hold for sufficiently large values of $n$?  

In short: yes.
The precise statement would be:

If the series with general term $b_n$ diverges, and there exists $N\geq 0$ such that $a_n \geq b_n \geq 0$ for all $n\geq N$, then the series with general term $a_n$ diverges.

(Similarly for the other way, to show convergence.)
Note: You can have more general statements involving more delicate conditions, but this is sufficient for most of the cases.
