Calculus of a Single Variable, Chapter 9, 9.3, Section 9.3, Problem 80

To evaluate the series sum_(n=2)^oo ln(n), we may apply the divergence test:
If lim_(n-gtoo) a_n != 0 then sum a_n diverges.
From the given series sum_(n=2)^oo ln(n) , we have a_n=ln(n) .
Applying the diveregence test,we determine the convergence and divergence of the series using the limit:
lim_(n-gtoo)ln(n) = oo
When the limit value (L) is oo then it satisfies lim_(n-gtoo) a_n != 0 .
Therefore, the series sum_(n=2)^oo ln(n) diverges.
We can also verify this with the graph: f(n) = ln(n) .

As the value of n increases, the function value also increases and does not approach any finite value of L.