1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

142 CHAPTER 4 • SEQUENCES, JULIA AND MANDELBROT SETS, AND POWER SERIES

00
t Corollary 4.2 If lzl > 1, the series I: z-n converges to f (z) = ,.: 1. That is,

n=l

if lzl > 1, then

00 1

I:

z - n = z - 1 + z -2 +... + z - n + ... = --,

z- 1

n=l

or equivalently,

00 1

-L...J'°"' z - n =-z -I -z -2 - ···- z -n - ···=

1

• z.
  n=l
  If I z I ::; 1, the series diverges.

P roof If we let ~ take the role of z in Equation ( 4-11), we get

00

(1)" 1

I:; = 1 -!

n=O z

if I ~I < 1.

Multiplying both sides of this equation by ~ gives

l

00

(1 )" 1

;I: ; = z-1

n=O

which, by Equation (4-10), is the same as

oo (!)n+l = _l

I: z z - 1

n=O

00

But this expression is equivalent to saying that :[ (~)" =. : 1 if 1 < lzl, which

n=l

is what the corollary claims.

It is left as an exercise to show that the series diverges if lz I ::; l.

t Corollary 4.3 If z :f. 1, then for all n ,

1 2 n - 1 z"

--=l+z+z +···+z +--.

1-z 1 - z

Proof This result follows immediately from Equation (4-14).

• EXAMPLE 4.13 Show that f ci;r = 1 -i.
  n =O
  Solution If we set z =^1 2;, then lzl = :/f' < 1. By Theorem 4.12, the sum is

__ 1~ = 2 = _2_ = 1 -i.

1-^1 - i 2 2-l+i l+i