Region of convergence for the series of functions Find out the region of convergence for the following series of function.
$$\sum_{m=1}^\infty x^{\log (m)}$$
Here $x \in \mathbb{R}$. This is a series of function. I was trying to find out the radius of convergence by Root Test and Ratio test. But no suitable solution I am getting.
What will be the region of convergence if we consider $x \in \mathbb{C}$?
Thank you for your help. 

One idea would be to use the fact that $\:x^{\log(m)}=e^{\log(m)\log(x)}=e^{\log(m)\Large(\normalsize\log(x)/\log(m)\Large)}\underset{m\to\infty}{\sim}e^{\log(m)}.$
