What are real world applications of power series ?
A few thoughts...
Power series are often used by calculators and computers to evaluate trigonometric, hyperbolic, exponential and logarithm functions. More accurately, a combination of power series and tables may be used in preference to the slower CORDIC algorithms used on more limited older hardware.
So any application of these kind of functions is a possible application of power series.
That includes all sorts of things from working out how much sand you will need for a triangular sandpit to astrophysics.