What are the values of r (with r>0) for which the series converges?
#r^ln(n)=n^ln(r)#. Then determine the values of r (with #r>0#) for which the series #sum_(n=1)^oor^ln(n)# converges.
I will just answer the part about the convergence, the first part having been answered in the comments. We can use
The series on the right is the series form for the famous Riemann Zeta function. It is well known that this series converges when
The result about the Riemann Zeta functions is very well known, If you want an ab initio answer, you can try the integral test for convergence .