Graphics Reference
In-Depth Information
The subsequence rule for null sequences
qn 36(a) Every subsequence of a null sequence is null.
The shift rule for null sequences
qn 36(c)
If for some fixed positive integer
k
,(
a
)isa
null sequence, then (
a
) is a null sequence.
If (
a
n
) and (
b
n
) are null sequences, then
The sum rule for null sequences
qn 44
b
(i) (
a
) is a null sequence;
The difference rule for null sequences
qn 45
(ii) (
a
b
) is a null sequence;
qn 45
(iii) (
c
·
a
d
·
b
) is a null
sequence;
The product rule for null sequences
qn 43
(iv) (
a
b
) is a null sequence;
The squeeze rule for null sequences
qn 46
(v) when
a
c
b
,
eventually, (
c
) is a null
sequence.
Convergent sequences and their limits
48 Examine the terms of the sequence given by
n
n
1
.
a
To what limit do you think this sequence tends?
What can you say about the sequence (
a
1)?
Up to this point we have avoided the term 'limit' because in
non-mathematical usage (
e
.
g
. a speed limit) it is synonymous with
'bound'. However, for mathematicians, bounds need not be approached,
whereas limits are numbers to which the terms of sequences and the
values of functions get close.
:
We will say that the sequence (
a
n
) tends to the limit
a
, or converges
to
a
, if and only if (
a
n
a
) is a null sequence.
This is expressed symbolically by writing
a
as
n
or
lim
a
.
either
(
a
)
a
We read each of the arrows here as
tends to
.
A sequence of numbers with a number as limit is said to be
convergent
.
