Quiz+2

Compute the value of the following sum: math \sum\limits_{n=1}^{\infty} n\left(\frac{1}{2i}\right)^n math (hint: first find a closed form for the sum math \sum_{n=1}^\infty n z^n math and determine whether this power series converges for math z=\frac{1}{2i} math there is no need to simplify your answer)

Find a harmonic conjugate of the function math f(x,y)=x^3-3xy^2 math

Let math S=\{z: |z|<1 \mbox{ or } |z-3|\leq 1\} math

1. Is the set S open, closed or neither?

2. What is its boundary math \partial S math ?

3. Is the boundary math \partial S math open, closed or neither?