Graphics Reference
In-Depth Information
The null sequence test
When discussing the series, there are two sequences, (
a
), which
we will look at. The relationship between these two sequences has one
straightforward aspect.
) and (
s
11 Suppose that the series
a
is convergent, and that its sequence of
s
as
n
s
s
partial sums (
s
)
. Usetheequation
a
and the difference rule, qn 3.54(v), to prove that (
a
) is a null
sequence.
So
a
convergent
(
a
) is null. So what happens if (
a
)is
not
null?
a
Clearly
cannot be convergent, if we wish to avoid a contradiction.
So wecan say (
a
) is not null
a
is divergent.
This
is the
null
sequence test
.
The argument here goes from (
P
Q
) to (not
Q
not
P
).
Repeating this argument gets you from (not
Q
not
P
)to(
P
Q
). So
(
P
Q
)
(not
Q
not
P
).
(not
Q
not
P
) is called the
contrapositive
of (
P
Q
).
12 By considering when the sequence (
x
)is
not
null, useqn 11 to
check your claims of divergence in qn 9.
13 Is there a positive real number
such that
n
is convergent?
Simple consequences of convergence
14 If the series
a
a
a
a
...
. . . is convergent to the sum
s
,
what can you say about the series
a
a
a
...
a
...?
Writeyour claim with the
notation. Writea proof taking
s
as
the
n
th partial sum of the first series and
t
as the
n
th partial sum
of the second. Now consider the converse problem. Suppose that
a
. . . is convergent to the sum
t
, what
can you say about the series
a
a
a
...
a
. . .? Write
your claim with the
notation and proveit. As a rsult of your
work, you should haveproved that
a
a
...
a
a
is convergent if and
only if
is convergent.
15 Provethat
a
is convergent if and only if
a
is convergent.
16
The start rule
Provethat
a
is convergent if and only if
a
is convergent.
This proves that
a
is not an ambiguous symbol if it is only
convergence which is at stake. It
is
an ambiguous symbol if thesum is
to befound.
