What is the radius of convergence for $\sum_{j=0}^{\infty}\frac{i^{j}+(-i)^{j}-i}{j!}z^{j}?$

Question

How do I find the radius of convergence for

\begin{equation} \sum_{j=0}^{\infty}\frac{i^{j}+(-i)^{j}-i}{j!}z^{j} \end{equation}

At first I thought I could do some simplification using the fact that $i^{3}=-i$, and then using the ratio test, but I wasn't able to get anywhere from there. Should I instead use some kind of bound?

Answer 1

You could notice that the modulus of the numerator is bounded between two constants, so applying the formula for the radius of convergence reduces to take the $n$-th root of $n!$ and evaluate the limit to $+\infty$.