Final answer:
Partial fraction decomposition involves breaking down a complex fraction into simpler ones that can be expanded into power series.
Step-by-step explanation:
To perform the partial fraction decomposition of a function h(z) and then expand it into a sum of two power series, we need to first decompose the function into simpler fractions that can be readily expanded.
The standard form of mathematical functions like trigonometric functions, exponential functions, and logarithms can be expressed as infinite sums. Similarly, to expand a fraction through partial fraction decomposition, we look for a common denominator and express each term in a form that allows us to work with a uniform denominator, analogous to how we find a common denominator when adding simple fractions like ½ and ⅓.
To find the coefficient for the zⁿ power term, we combine the terms of the separate power series after expansion. The series expansions can be calculated using methods such as the binomial theorem or by performing basic operations like multiplication of exponentials (add the exponents) or division of exponentials (subtract the exponents).
Typically, a closed-form expression is not possible for the coefficient of the zⁿ term, but in certain cases such as the Fibonacci sequence, recognizing patterns can lead us to a solution.