Graphics Reference
In-Depth Information
Now supposethat thefirst
N
terms of the sequence of
a
s are
contained within the first
L
terms of the sequence of
b
s. Provethat
s
B
. Deduce that
A
B
, so that
A
B
.
This establishes that when a convergent series of
positive
terms is
rearranged it is always convergent to the same sum.
t
77 Let
a
be an absolutely convergent series and let
b
bea
rearrangement of the same series.
As in qn 67, welt
u
(
a
a
),
v
(
a
a
),
x
(
b
b
),
y
(
b
b
).
Why are the two series
u
necessarily convergent?
Are they both sequences of positive or zero terms?
How does
u
and
v
relate to
x
, and
v
to
y
?
a
v
y
b
Provethat
.
This establishes that when an
absolutely convergent
series is
rearranged, it is always convergent to the same sum.
(
u
)
(
x
)
Summary
-
series of positive and negative terms
Alternating series test
qn 63
If (
a
) is a monotonic null sequence then
(
1)
a
is convergent.
Definition
If
a
is convergent, then
a
is said to be
absolutely convergent
.
Absolute convergence test
qn 67
If
a
is absolutely convergent then
a
is
convergent.
Definition
A series
which is convergent, but not
absolutely convergent, is said to be
conditionally
convergent
.
a
Rearrangements
qns 74, 76
If the terms of a conditionally convergent series
are rearranged, the sum of the series may be
changed, or the rearranged series may diverge.
Theorem
qn 77
If the terms of an absolutely convergent series
are rearranged, the rearranged series has the
samesum.
