Final answer:
A power series is a series of the form Σn=0∞ an(z - z0)n, where z0 is the center, and an are the coefficients. For a power series centered at 3i with a radius of convergence 7, a choice of coefficients could be (-1)n/n!. For a power series centered at 1 and convergent for all complex numbers, a choice of coefficients could be 1 for all n.
Step-by-step explanation:
A power series is a series of the form Σn=0∞ an(z - z0)n, where z0 is the center, an are the coefficients, and z is a complex number. Let's consider the properties mentioned:
a) To find a power series centered at z0 = 3i with a radius of convergence R = 7, we can choose any coefficients an. For example, an = (-1)n / n! would give a power series: Σn=0∞ (-1)n(z - 3i)n.
b) To find a power series centered at z0 = 1 and convergent for all complex numbers, we can choose the coefficients such that the power series converges everywhere. For example, an = 1 for all n would give a power series: Σn=0∞ (z - 1)n.