1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

4.3 • GEOMETRIC $ERIES AND CONVERGENCE THEOREMS 1 4 5

in the sequence larger than L +t:, b ecause L+t: < 3, as the following calculation
shows:

• EXAMPLE 4.19 The limit supremum of the Fibonacci sequence

{ t.,} = {l , 1, 2, 3, 5, 8, 13, 21, 34, ... } is lim sup tn = oo.

n -oo

(The Fibonacci sequence satisfies the relation tn = tn-I + tn-z for n > 2.)

The limit supremum is a. powerful idea. because the limit supremum of a. se-

quence always exists, which is not true for the ordinary limit. However, Example

4.20 illustrates the fa.ct that, if the limit of a. sequence does exist, then it will be

the same a.s the limit supremum.

• EXAMPLE 4.20 The sequence

{tn} = {1+~}

= { 2, 1.5, 1.33, 1.25, 1.2, ... } has Jim sup tn = 1.

n-+oo

We leave verification of this as an exercise.