Find the radius of convergence for $\sum_{k \geq 0} \cos(k)z^k$.
I was trying to use the root test or ratio test, but the only thing I know is $$\lim_{k \rightarrow \infty}[\cos(k)]^{1/k}\leq 1$$ And I can only say the radius of convergence $R \geq 1$. Can anyone give a more rigorous result? Thanks!
$$\sum_{k=0}^{\infty}\cos (k)z^k=\sum_{k=0}^{\infty}\frac{e^{ik}+e^{-ik}}{2}z^k$$so the convergence radius is where $|ze^i|<1$ and $|ze^{-i}|<1$ which leads to $|z|<1$ and $R=1$