154 CHAPTER 4 • SEQUENCES, JULIA AND MANDELBROT SETS, AND POWER SERIES
00
(i) g (z) = L: z^2 ".
n=O
00 n
(j) g (z) = L: !!..,z". Hint: lim [1 + ( .! )]" = e.
tt·=O n. n-oo n
00- Show that L: (n + 1)^2 z" = ci1!rj 3. For what values of z is this valid?
n=O
00 00 - Suppose that L: c,,z" has radius of convergence R. Show that L: c;.z" has radius
n=O n = O
of convergence R^2 •
00 - Does there exist a power series I: c,,z" that converges at z 1 = 4 - i and diverges
n::O
at z2 = 2 + 3i? Why or why not? - Verify part (ii) of Theorem 4.17 for all k by using mathematical induction.
- This exercise establishes that the radius of convergence for g given in Theorem
4.17 is p, the same as that of the function f.
(a) Explain why the radius of convergence for g is --~^1 ~-~
nUm -oo suptnc:.,.I~.
(b) Show that Lim sup nn:!:T = 1. Hint: The lim sup equals the limit.
n-oo
Show that n -lim oo !2&.!!. n - 1 = 0.
(c) Assuming that lim suplc,.j~ = Jim suplenl*, show that the con-
n-oo n-oo
clusion for this exercise follows.
(d) Verify the truth of the assumption made in part (c).- Here we establish the validity of Inequality (4-17) in the proof of Theorem 4.17.
(a) Show that
I
s"_ t" I =
s-t
lsn- 1 + s"-2t + s"-3t2 + ... + stn-2 + t"-' I
~ ls"-
1
I + ls"-
2
tl + ls"-^3 t
2
I + · · · + lst"-
2
1 + It"-' I,
where s and t are arbitrary complex numbers, s ¥-t.
(b) Explain why, in Inequality (4-17), lz -a:I <rand lzo -ad < r.
(c) Lets= z - a and t = zo - a in part (a) to establish Inequality (4-17).10. Show that t he radius of convergence of the series for Jo (z) and J 0 (z) in Example
4.25 is infinity.
1 1. Consider the series obtained by substituting for the complex number z the real
number x in the Maclaurin series for sin x. Where does this series converge?(^12). Sh ow th a , t 1 ' or I z -t ·1 < v r.:2 .<, r::z I = "00 Lm=O (l-i)(z- i")" +' ·
Hint: 1 ~, = (l-i)~(z-i) = ,:. [ 1 _ ~-'.]. Now use Theorem 4.12.
1 -•