For f holomorphic near a with f'''(a) = 18, what is the coefficient of (z - a)³ in its Taylor series about a?

The series sum over n ≥ 0 of n! z^n has coefficients growing without bound. What is its radius of convergence?

• 0
• 1
• Infinite

The series 1 + z + z² + ... sums to 1/(1 - z). For which z does it converge?

• Everywhere in the plane
• Inside the unit disc, |z| < 1
• Only at zero itself

The radius is 1 because the function 1/(1 + x²) blows up at the real point x = 1.

On |z| ≤ 1/2, what is the minimum truncation degree N guaranteeing the geometric tail is at most 1/8 in modulus?

The function 1/(1 + z²) is expanded about 1. What sets its radius of convergence?

• Uncancelled poles at ±i, each at distance √2 from 1
• Poles at ±i, each at distance 1 from the centre
• Real denominator zeros at ±1, at distance 0

For the series sum over n ≥ 1 of z^n/n, which boundary point converges?

• z = 1
• z = -1
• Every point of |z| = 1

A function analytic on the union of disjoint disks about 0 and 3, equal to 0 on the first and 1 on the second, shows that zeros accumulating in one component do not force the function to zero on the whole union. True or false?

f and g are analytic on the unit disk with f(1/n) = g(1/n) for every integer n ≥ 2, yet f'(0) = g'(0) + 1. Is this consistent?

f is analytic on the unit disk with f(1/n) = 1/n² for every integer n ≥ 2. What is f throughout the disk, and what is the order of its zero at 0?

• f(z) = z², zero of order 2
• f(z) = z², zero of order 1
• Only the values on the sequence are known