Writing a complex function as a power series?

Question

I have been asked to write the following summation as a power series:

$$\sum_{n \geq 500} i^n \frac {z^{5n-2}}{n!}. $$

I know that by comparison to the power series $$\sum_{n \geq 0} a_n (z-a)^n, $$ we can let $$ a_n = \frac {i^n}{n!},$$ and we can let $$a=0.$$ I am unsure how to represent the $$ n \geq 500 \text { and the power } 5n-2. $$

Thanks

Answer 1

Define$$a_n=\begin{cases}\frac{i^{\frac{n+2}5}}{\left(\frac{n+2}5\right)!}&\text{ if }5\mid n+2\text{ and }n\geqslant2\,498\\0&\text{ otherwise.}\end{cases}$$