Graphics Reference
In-Depth Information
Summary
-
convergent sequences
Definition
A sequence (
a
) is said to tend to a limit
a
or to
be convergent, if any only if, given
0, there
exists an
N
, such that
n
N
a
a
.
This is expressed symbolically by writing
(
a
)
a
as
n
.
The subsequence rule
qn 54(ii)
If (
a
)
a
then every subsequence of (
a
)
a
.
The shift rule
qn 52
If, for some fixed integer
k
,(
a
)
a
, then
(
a
)
a
.
The scalar rule
qn 54(i)
If (
a
)
a
, then (
c
·
a
)
c
·
a
.
The absolute value rule
qn 54(vii)
If (
a
)
a
, then (
a
)
a
.
The reciprocal rule for non
-
null sequences
qns 64, 65
If (
a
)
a
, and
a
,
a
0, then (1/
a
)
1/
a
.
Theorem
qn 62
Every convergent sequence is bounded.
The closed interval rule
qn 78
If (
a
)
a
and
A
a
B
, then
A
a
B
.
If (
a
n
)
a
and (
b
n
)
b
then
The sum rule
qn 54(iii)
(i) (
a
b
)
a
b
;
T
he difference rule
qn 54(v)
(ii) (
a
b
)
a
b
;
qn 54(iv)
(iii) (
c
·
a
d
·
b
)
c
·
a
d
·
b
;
The product rule
qn 54(vi)
(iv) (
a
b
)
ab
;
The squeeze rule
qn 54(viii)
(v) when
a
c
b
, eventually, and
a
b
,
a
;
(
c
)
The quotient rule
qn 67
(vi) when
b
,
b
0, (
a
/
b
)
a
/
b
;
The inequality rule
qn 76
(vii) when
a
b
,
a
b
.
Theorem
qn 51
Every real number is the limit of some sequence
of terminating decimals.
If 0
a
,(
a
)
1.
Theorem
qns 57, 68(x)
(
n
)
Theorem
qn 59
1.
