4 .4 • POWER SERIES FUNCTI ONS 147
Est ablish t he claim in the proof of Theorem 4.12 that if lzl < 1, then n -Jim oo z" = O.
In the geometric series, show that if lzl > 1, t hen n-oo lim I Sn I = oo.
Prove that the series in Corollary 4.2 diverges if lzl :5 1.
Prove Theorem 4.13.
Give a rigorous argument to show that lim supt., = 1 in Example 4.20.
n-oo
00 ( n n
For lzl < 1, let f (z) = E z^2 > = z + z^2 + z^4 + · · · + z<^2 > + · · ·. Show that
n::. 0
f (z) = z + f (z^2 ).
This exercise makes interesting use of the geometric series.
(a) Use the formula for geometric series with z = re'^9 , where r < 1, to
show thatL
oo n Loo n in9 1 - rcosB+irsinB
z = r e =.
n=O n=O 1 + r^2 - 2r cos 8(b) Use part (a) to obtainL
oo 1 -rcosBr"cosnB = ---.,,---..,.
n=O^1 + rZ -2rcos8
(^00) r~nB
°'"' L..J r"sinnB = -1 + --=rZ -- --2rcos8..,. ·
n::O
and
4.4 Power Series Functions
00Suppose that we have a series L: (n, where (n = Cn (z - a)". If °' a nd the
n=O
collection of Cn are fixed complex numbers, we get different series by selecting
different values for z. For example, if °' = 2 and en = ~ for all n, we get
00 00
the series L: ~ (~ -2t if z = ~.and L: ~ (2 + i)" if z = 4 + i. Note that,
n=O n=O
when °' = 0 and c,, = 1 for all n , we get the geometric series. T he collection of
00
points for which the series L: en (z - a)" converges is the domain of a function
n=O
00
f (z) = L: en (z -a)", which we call a p owe r series function. Technically,
n = O
this series is undefined if z = a and n = 0 because o^0 is undefined. We get around
00
this difficulty by stipulating that the series L: Cn (z - a)" is really compact
n=O
00
notation for Co+ L: en (z - a)". In this section we present some results that
n=I
are useful in helping establish properties of functions defined by power series.