Okay, this is the editied version.
Another old exam question that I can't shake from my mind. I am searching for an intermediate form (most probably the Taylor series) such that it is possible to analytically continue the function $$f(z) = \sum_{n=0}^\infty \frac{z^n}{n+\frac{1}{2}}$$ outside of the unit circle. The next step will be to write the function $f\,$ in a hypergeometric form. Any thoughts or insight would be greatly appreciated.
Thanks