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Wednesday, February 8
−
Lecture 19 :
Tests for convergence of series : Limit
Comparison test.
(Refers to Section 8.4 in your text)
After having practiced the problems associated to the concepts of this lecture the student should
be able to
:
Apply the
Limit comparison test
to determine convergence or
divergence of a series.
19.1
Theorem
The Limit Comparison test
.
Suppose {
a
j
:
j
= 1, 2, 3, …} and
{
b
j
:
j
= 1, 2, 3, …} are both sequences of
nonnegative
terms. Then:
Proof:
1.
Suppose lim
j
→ ∞
(
a
j
/
b
j
) =
L
> 0,
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View Full Document 2.
Suppose lim
j
→ ∞
(
a
j
/
b
j
) = 0,
(For part 3: Remember
lim
j
→
∞
c
j
=
∞
means that for any integer
N
> 0 no matter how
large, the interval
[
N
,
∞
)
contains a tail end of the sequence {
c
j
}.)
3.
Suppose lim
j
→ ∞
(
a
j
/
b
j
) =
∞
. Let
M
be any positive integer.
19.2
Examples
a
)
Test for convergence the series
Solution:
Let
Observe that when
n
is large
a
n
only slightly distinguishes itself from
b
n
= 3/
n
2
.
Hence if we ignore the terms 1/
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This note was uploaded on 03/19/2012 for the course MATH 118 taught by Professor Zhou during the Winter '08 term at Waterloo.
 Winter '08
 ZHOU
 Calculus

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