1600 CHAPTER 49. THE COMPLEX NUMBERS

49.2 Exercises1. Prove the root test for series of complex numbers. If ak ∈ C and

r ≡ lim supn→∞

|an|1/n

then∞

∑k=0

ak

 converges absolutely if r < 1diverges if r > 1test fails if r = 1.

2. Does limn→∞ n( 2+i

3

)nexist? Tell why and find the limit if it does exist.

3. Let A0 = 0 and let An ≡ ∑nk=1 ak if n > 0. Prove the partial summation formula,

q

∑k=p

akbk = Aqbq−Ap−1bp +q−1

∑k=p

Ak (bk−bk+1) .

Now using this formula, suppose {bn} is a sequence of real numbers which convergesto 0 and is decreasing. Determine those values of ω such that |ω|= 1 and ∑

∞k=1 bkωk

converges.

4. Let f : U ⊆ C→ C be given by f (x+ iy) = u(x,y)+ iv(x,y) . Show f is continuouson U if and only if u : U → R and v : U → R are both continuous.