Calculus of a Single Variable, Chapter 9, 9.2, Section 9.2, Problem 49

sum_(n=2)^oo n/(ln (n))
To determine if the series is convergent or divergent, apply the nth-Term Test for Divergence.
It states that if the limit of a_n is not zero, or does not exist, then the sum diverges.

lim_(n->oo) a_n!=0 or lim_(n->oo) a_n =DNE
:. sum a_n diverges

Applying this, the limit of the term of the series as n approaches infinity is:
lim_(n->oo) a_n
=lim_(n->oo) n/ln(n)
To take the limit of this, use L’Hospital’s Rule.
=lim_(n->oo) (1)/(1/n)
=lim_(n->oo) n
=oo
Therefore, by the nth-Term Test for Divergence, the series diverges.