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Monday, January 30
−
Lecture 14 :
Monotone convergence (sequence) theorem
(Refers to Section 8.1 in your text)
After having practiced the problems associated to the concepts of this lecture the student should
be able to
:
State the
Monotone sequence theorem
(
Monotone convergence theorem
), apply the
Monotone convergence theorem to determine convergence of a sequence.
If we know a sequence converges but cannot compute its limit we can always
approximate its limit by choosing a term in its “tailend. A rough estimate of its level of
accuracy can be obtained by comparing the value of a few terms around the term we have
chosen. But before we approximate the limit of sequence we must determine whether the
sequence converges at all. The
Monotone sequence theorem
provides a tool to do this.
14.1
Definitions
A sequence of numbers {
a
i
:
i
= 1, 2, 3,.
..} is said to be
increasing
if
a
i+
1
>
a
i
for all
i
≥
1.
decreasing
if
a
i+
1
<
a
i
for all
i
≥
1.
nonincreasing
if
a
i+
1
≤
a
i
for all
i
≥
1.
nondecreasing
if
a
i+
1
≥
a
i
for all
i
≥
1.
A sequence of numbers is said to be
monotonic
or
monotone
if it is either
increasing, decreasing, nonincreasing or nondecreasing.
14.2
Definitions
Let
S
= {
a
i
:
i
= 1, 2, 3,.
..} be a sequence of numbers.
We say that a number
M
is an
upper bound
of
S
if
a
i
≤
Μ
for all
i
.
We say that a number
L
is a
lower bound
of
S
is
L
≤
a
i
for all
i
.
If a sequence has both an upper bound and a lower bound then we say it is
bounded
.
If a sequence has no upper (lower) bound
we say that the sequence is
unbounded above
(
below
).
The smallest upper bound of a sequence is called the "
least upper bound
" (or
supremum
) and the largest upper bound of a sequence is called the "
greatest
lower bound
" (or
infimum
).
Note that these definitions apply to arbitrary subsets of
R
as well as sequences.
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View Full Document14.3
The Monotone sequence theorem
(
Monotone convergence theorem
)
−
Any
bounded monotonic sequence must converge to some number.
Proof: (
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 Winter '08
 ZHOU
 Calculus

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