Convergence for power series $\sum_{n=0}^{\infty}z^n(5+e^{i\pi n})^n$? Consider the power series
$$\sum_{n=0}^{\infty}z^n(5+e^{i\pi n})^n \tag{1}.$$
Identifiying $e^{i\pi n}$ as $(-1)^n$ for all integers $n$, highlights the problem with using the ratio test or the root test for convergence of series, since $(-1)^n$ does not converge, right?
How can the convergence be determined for this power series? Does it exist?
Does it exist for other than $z=0$? Can it be determined by setting $(-1)^n=-1$ for all $n$ and therefore calculate the "smallest" radius or something like that? 

Yes, $e^{i\pi n}=(-1)^n$. Therefore, your series is$$6+4z+6^2z^2+4^3z^3+\cdots$$And, yes, you can use the root test here:$$\limsup_n\sqrt[n]{\left\lvert z^n\bigl(5+(-1)^n\bigr)^n\right\rvert}=6\lvert z\rvert.$$Therefore, the radius of convergence is $\frac16$.
