Final answer:
The question is about the series representations of functions, convergence of power series, and the properties of series with complex number bases in mathematics.
Step-by-step explanation:
The student's question involves series representations of functions, specifically power series, and their region of convergence. The function f₁(z) is defined as a power series with a complex number base (1+i) raised to the power of n, and the function f₂(z) is a power series with coefficients aₙ and the variable z shifted by 1. The region of convergence refers to the set of values for z where the power series converges to a function.
Power series can represent many standard mathematical functions and have properties that allow for analysis and manipulation using various mathematical techniques. The question appears to be part of a larger discussion on power series expansions and their applications.