The partial sum 1 + 0.5 + 0.25 + 0.125 approximates 1 divided by (1 minus 0.5). What is the value of this partial sum?

Answer: ______________

The geometric series 1 + z + z squared + ... sums to 1 divided by (1 minus z). For which z does it converge?

• For every complex z
• For |z| below 1
• For |z| above 1
• Only for z equal 0

The series 1 plus z plus z squared and on sums to 1 over (1 minus z). For which z does it converge?

• Everywhere in the plane
• Inside the unit disc, modulus below 1
• Only at zero itself

The radius is 1 because the function 1 over (1 plus x squared) blows up at the real point x equal 1.

Circle one:   True   False

The geometric series for 1 divided by (1 minus z) is truncated after the z cubed term. At z equal 0.5 the discarded tail is 0.5 to the power 4 plus 0.5 to the power 5 and so on. What is the tail sum?

Answer: ______________

Why does the real Taylor series of 1 over (1 plus x squared) about 0 stop at radius 1?

• Real series always stop at 1
• The function breaks at x equal 1
• Complex poles one unit away cap it

On which disc is the Taylor series of 1 divided by (1 minus z) about 0 valid?

• The disc |z| below 1
• The whole complex plane
• The disc |z| below 2
• The region |z| above 1

A nonzero analytic function on a domain can have zeros that accumulate at a point inside the domain. True or false?

Circle one:   True   False

A nonzero analytic function on a domain can have zeros accumulating at an interior point.

Circle one:   True   False

Which zero set can belong to a nonzero function analytic on the whole complex plane?

• Every point of the interval [0, 1]
• A sequence of distinct points converging to 0
• The points k times pi for every integer k
• Every rational number