Math 534 HOMEWORK 1 (due Wednesday, Oct. 3) Autumn 2007
1. (a) Let |a| ≤ 1 and |b|<1. Show
a−b
1−ab
≤1.
When does equality occur?
(b) Calculate (±1±i√3
2)6for all combinations of signs.
2. Let a=e2πiα with α6∈ Q. Prove that the sequence {an}∞
n=0 is dense in the unit circle
T={z:|z|= 1}.
3. Let {an}be a sequence of complex numbers with the three properties:
(i) Re(an)≥0 for all n,
(ii) P∞
n=1 anconverges,
(iii) P∞
n=1 a2
nconverges.
Prove that P∞
n=1 |an|2converges. Show in addition that none of the three assumptions can
be omitted.
4. Let anbe a sequence of complex numbers such that the sequence sn=a0+a1+· · · +an
is bounded. Let bnbe a decreasing sequence of real numbers with limn→∞ bn= 0. Prove that
P∞
n=0 anbnconverges.
5. For what values of zis P∞
n=0(z
1+z)nconvergent? Draw the picture of the region.
6. Find the radius of convergence Rof P∞
n=0 z2nand show that the function represented by
this series has no continuous extension to any arc of the circle {z:|z|=R}.
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