126 CHAPTER 4 • SEQUENCES, JULIA AND MANDELBROT SETS, AND POWER SERIES
We needed to show the strict inequality lzn -(I < £, and the next-to-last
line in the proof gives us precisely that. Note also that we have been speaking
of the limit of a sequence. Strictly speaking, we are not entitled to use this
terminology because we haven't proved that a complex sequence can have only
one limit. The proof, however, is almost identical to the corresponding result for
real sequences, and we leave it as an exercise.
- EXAMPLE 4. 1 Find Jim Zn if Zn= fo+i(n+I).
n~oo n
S o u I tion • UT vve wn •te Zn - Xn + iyn · - fo I + i..,,. •n+ l U smg • res uJt s concernmg •
sequences of real numbers, we find that n-oo Jim Xn = n-oo Jim vn ~ = 0 and n - oo lim Yn =
lim !!±.! = 1. Therefore, lim Zn = lim Yfi+i(n+l) = i.
n-oo n n-oo n-oo n•EXAMPLE 4.2 Show that {(l +i)"} diverges.
Solution We have
Zn = (1 + i)" = (hf cos :
11
+ i ( h)" sin n
4n.The real sequences { ( v'2) n cos n 4 "} and { ( h) n sin ".i"} both diverge, so we
conclude that { (1 + i}"} diverges.
Definition 4.2: Bounded sequence
A complex sequence {zn} is bounded provided that there exist a positive real
number R and an integer N such that lznl < R for all n > N. In other words,
for n > N, the sequence {Zn} is contained in the disk DR (0).Bounded sequences play an important role in some newer developments in
complex analysis that are discussed in Section 4.2. A theorem from real analysis