1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences and Series 209

5. Consider the series

2

2 1 2 1 3 1 3m

+ 3 Z + 3 Z + 22. 3 z + ... + 2sm-13m z

1 z3m+l 1 z3m+2

+ 2sm-l3m+i + 2sm3m+1 + · · ·

Here

I a:::l I =

3

' I ::::~ I =

2

' I ::::: I =

4

Hence, by (4.11-10), R = ij3.2.4 = 2-ij3. Otherwise: As m-+ oo, we obtain

1

-+ --

2-ij3'

3m+v -----+ (^1) -- 1
2sm-13m+1 2-ij3'
Hence, by (4.11-4), R = 2-ij3.

6. Consider the series 2=: CnZn, where {en} is the Fibonacci sequence,

defined by co = 0, c1 = 1, and Cn = Cn-l + Cn-2 for n ~ 2. It can be shown
by mathematical induction that Cn = (an -(3n )/VS, where a =^1 / 2 (1 +VS)

and (3 =^1 / 2 (1 - VS). Hence

I

Cn I an-(3n l-((3/a)n

Cn+l = an+l - (3n+l = a - ((3/a)n(3

But lf3/al = (VS - 1)/( VS+ 1) < 1. Thus ((3/a)n -+ 0 as n-+ oo, and
we get

R=~=%(v'5-1)

a

Note As stated before, a series of the form

can be written as 2:: anz^1 n by letting z-a 0 = z^1 • If the disk of convergence
of the last series is lz^1 I < R, then the region of convergence of the given
series is the disk lz - zo I < R.

Theorem 4.23 (Identity Principle for Power Series) Let f(z) = 2=: anzn

for lzl < Ri and g(z) = 2:: bnzn for lzl < R2· If f(z) = g(z) for all

z E N 0 (0), where 0 < 5 < min(R 1 , R2), or if J(zk) = g(zk), where {zk} is

a sequence of points in N5(0) such that zk -+ 0 as n -+ oo, then an = bn