Anyone know what they did in this answer key here? like how did they changed the interval of the series from k=-m-> m, to k=1-> m. and how they got z^k to the step with the (*).

Water in a partially filled large tank is to be supplied to the roof top, which is 8 m above the water level in the tank, through a 2.2-cm-internal-diameter pipe by maintaining a constant air pressure of 300 kpa (gage) in the tank. if the head loss in the piping is 2 m of water, determine the discharge rate of the supply of water to the roof top in liters per second.