Further Notes on Sequences
Additional Methods of Determining Convergence/Divergence of a
Sequence
:
Using rules to evaluate limits
: Just as there are rules for evaluating limits
of functions, there are similar rules for evaluating limits of sequences.
They are listed below.
Suppose
{
}
n
a
and
{
}
n
b
are convergent sequences and
c
is a constant. Then
1.
(
29
lim
lim
lim
n
n
n
n
n
n
n
a
b
a
b
→∞
→∞
→∞
±
=
±
2.
(
29
lim
lim
lim
n
n
n
n
n
n
n
a b
a
b
→∞
→∞
→∞
=
3.
lim
n
c
c
→∞
=
;
(
29
lim
lim
n
n
n
n
ca
c
a
→∞
→∞
=
4.
lim
lim
lim
n
n
n
n
n
n
n
a
a
b
b
→∞
→∞
→∞
=
, provided
lim
0
n
n
b
→∞
≠
5.
(
29
lim
lim
p
p
n
n
n
n
a
a
→∞
→∞
=
6.
If
n
n
a
b
≤
for all but a finite number of terms, then
lim
lim
n
n
n
n
a
b
→∞
→∞
≤
Item 5 above holds provided the symbols make sense. For example, one
cannot take the square root of negative numbers.
Example: Below is an example in which this method is used to determine
convergence or divergence.
E1)
Consider
2
1
2
4
n
n
n
∞
=
+
.
2
2
2
2
2
2
2
2
lim
lim
lim
4
4
4
1
n
n
n
n
n
n
n
n
n
n
n


→∞
→∞
→∞
=
⋅
=
+
+
+
(Algebra)
2
2
lim
4
lim 1
n
n
n
n
→∞
→∞
=
+
2
1
2lim
1
lim1
4lim
n
n
n
n
n
→∞
→∞
→∞
=
+
(#4, and #3, #1)
2
1
2lim
1
lim1
4 lim
n
n
n
n
n
→∞
→∞
→∞
=
+
(#5)
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(
29
2
2 0
0
1
4 0
⋅
=
=
+
.
(Evaluating the limits)
Comparing with other sequences
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 Spring '11
 BUEKMAN
 Addition, Squeeze Theorem, Limits, lim, Mathematical analysis, Limit of a sequence

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