1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

4.4 • POWER $ERIES FUNCTIONS 149

y

Divergence

What happens on the

boundary may be unknown.

Figure 4.3 The radius of convergence of a power series.

We now give an example illustrating each of these cases.

• EXAMPLE 4.21 The series n~O ( ;,=;.^21 ) n (z -4)"' has radius of convergence

3 by Cauchy's root test because lirn lenl ~ = lim 3 'j.^21 = ~·
n-oo n-+oo

00

•EXAMPLE 4. 22 The series I: enz"' =4z+5^2 z^2 + 43z3+54z^4 +45z5+· · ·

n=l

has radius of convergence! by the Cauchy- Hadamard formula. We see this

result by calculating {lenl; } = {4, 5, 4, 5, ... }, so lim sup lenl; = 5.

n--+oo

00

• EXAMPLE 4.23 The series I: ;hz" has radius of convergence oo by the ratio
  n=O
  test because n - oo lim I (n~'t)· ' I = n- oo lim In+\ I = 0.