1. ## Domain of Convergence

I am given the problem:

Show that $r^{ln(n)} = n^{ln(r)}$ and determine the values of r $(r > 0)$ for which the series $\sum_{n=1}^{\infty} r^{ln(n)}$ converges.

I guess my initial problem is to show that the two are equal to one another as such. However, I thought that the minimum value of r that would converge would be 0 and the max would be 1.

If someone could just walk through the initial part of:

$r^{ln(n)} = n^{ln(r)}$

I would be very greatful. Thanks guys.

I am given the problem:

Show that $r^{ln(n)} = n^{ln(r)}$ and determine the values of r $(r > 0)$ for which the series $\sum_{n=1}^{\infty} r^{ln(n)}$ converges.

I guess my initial problem is to show that the two are equal to one another as such. However, I thought that the minimum value of r that would converge would be 0 and the max would be 1.

If someone could just walk through the initial part of:

$r^{ln(n)} = n^{ln(r)}$

I would be very greatful. Thanks guys.

It follows at once from the very definition of logarithms that $a^b=e^{b\ln a}\,,\,\,a>0$ . Now apply this to both sides of the given equality and verify it is really an equality.

Tonio