Lecture 15: Sequences (II)
Monotonic Sequence Thmorem
A sequence is
increasing
if
a
n
< a
n
+1
;
decreasing
if
a
n
> a
n
+1
:
A sequence is
mono-
tonic
if it is either increasing or decreasing.
A sequence is
bounded above
if there is a
number
M
such that
a
n
M
for all
n
1;
bounded below
if there is a number
m
such
that
m
a
n
for all
n
1.
A sequence is
bounded
if it’s bounded above and below.
Monotonic Sequence Thm:
Every bounded,
monotonic sequence is convergent.
Alternative statements of the theorem:
If a sequence
f
a
n
g
is monotonically increasing
and is bounded above, then
f
a
n
g
is convergent.
If a sequence
f
a
n
g
is monotonically decreasing
and is bounded below, then
f
a
n
g
is convergent.
1

ex.
f
a
n
g
is a monotonic increasing sequence such
that
a
1
= 2
; a
n
+1
=
1
2
(
a
n
+ 6)
for
n
2
:
and
a
n
<
6 for all
n
. Find the limit of the sequence.
ex.
f
a
n
g
is a monotonically decreasing sequence
such that
a
1
= 2
; a
n
+1
=
1
3
a
n
for
n
1
:
and
< a
n
2 for all
n
.
Find the limit of the sequence.
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