First lets start with a trick to simplify the right side. Multiply the equation through by 1/z and then take the difference of the two equations. That yields
iz^2 - iz = 1 + 1/z + 1/z^2 + 1/z^3 + 1/z^4 + ...
The right side is now a geometric series. Then
iz^2 - iz = 1/(1 - 1/z)
Multiply both sides by 1/z and then simplify a bit to get
Solution