1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

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

• EXAMPLE 4.16 Show that the series f: (z;:,!)" converges for all values of

n=O
z in the disk lz - ii < 2 and diverges if lz - ii > 2.

Solution Using the ratio test, we find that

If lz - ii < 2, then L < 1, and the series converges. If lz - ii > 2, then L > 1,
and the series diverges.

Our next result, known as the root test, is slightly more powerful than the

ratio test. Before we present this test, we need to discuss a rather sophisticated

idea used with it-the limit supremum.

Definition 4.10: Limit supremum

Let {tn} be a sequence of positive real numbers. The limit supremum of the

sequence (denoted lim suptn) is the smallest real number L having the property

n-oo

that for any e > 0, there are at most finitely many terms in the sequence that

are larger than L + e. If there is no such number L, then Jim sup tn = oo.

n-oo

• EXAMPLE 4. 17 The limit supremum of the sequence

{tn} = {4.1, 5.1, 4.01, 5.01, 4.001, 5.001, ... } is n-+oo lim sup tn = 5,

because if we set L = 5, then for any e > 0, there a.re only finitely many terms

in the sequence larger than L + e = 5 + e. Additionally, if L is smaller than 5,

then by setting e = 5 - L, we can find infinitely many terms in the sequence

larger than L + e (because L + e = 5).

• EXAMPLE 4.18 The limit supremum of the sequence

{tn} = {1, 2, 3, 1, 2, 3, 1, 2, 3, l, 2, 3, ... } is lim sup tn = 3,

n-oo

because if we set L = 3, then for any e > 0, there are only finitely many terms

(actually, there are none) in the sequence larger than L +e = 3+e-. Additionally,

if Lis smaller than 3, then by setting e =^3 2L we can find infinitely many terms