Holomorphic Extension

Hello. I've been trying to show that the function



cannot be holomorphically extended outside the unit disk . I'd like to show that the function "blows up" at the boundary, but it isn't clear to me how to do that, except for simple points like

I'd appreciate any suggestions you may have.

take z to be a point on the unit circle with rational argument, i.e. z=exp(ik/m). then eventually all those terms become 1/n...... u can do the rest :)