# What are the values of r (with r>0) for which the series converges?

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Show that #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.

Show that

##### 1 Answer

#### Explanation:

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 .