Graphics Reference
In-Depth Information
a
t
This follows from
s
.
Theequation
s
a
t
guarantees that (
s
) has a limit if and only
if (
t
) has a limit.
15 Let
u
bethepartial sum of thefirst
n
terms of
a
, then
s
a
a
u
. Now (
s
) has a limit if and only if (
u
) has a limit.
16 Let
v
bethepartial sum of thefirst
n
terms of
a
, then
s
a
a
...
a
v
. Now (
s
) has a limit if and only if (
v
)
has a limit.
17
s
s
. This tends to 0 by definition.
0, there exists an
N
such that
n
N
s
s
18 Given
or
a
.
19 If
s
is the
n
th partial sum of
a
, then
c
·
a
is the
n
th partial sum of
c
·
a
. By 3.54(i), thescalar rul, (
s
) is convergent if and only if (
c
·
s
)
is convergent, provided
c
0.
20 When
c
0,
c
·
x
is convergent if and only if
x
1.
21 If the
n
th partial sum of
a
is
s
and the
n
th partial sum of
b
is
t
,
and (
s
)
s
,(
t
)
t
, then (
s
t
)
s
t
, by thesum rul, 3.54(iii). But
s
t
is the
n
th partial sum of
(
a
b
).
22
(a) Yes. Unless the subscript is given to be constant, it is assumed to
bea variabletaking thevalus 1, 2, 3, . . ..
(b) (i) and (ii) only.
23
n
3. Useqn 4.35 for the
convergence of monotonic bounded sequences.
2
n
!
2
1/
n
!
1/2
e
s
24 Since2
e
3, e
p
/
q
q
3.
p
/
q
n
/
q
!
k
/
q
! for some integer
k
.
1/(
q
1)!
1/(
q
2)!
...
(1/
q
!)(1/3
1/3
...)
/
q
! which
contradicts the previous line. The proof of this result is set out very
fully in Bryant, pp. 20
—
2.
25
s
) is increasing it is convergent if and
only if it is bounded by qn 4.35.
s
a
s
. Since(
s
26
Let
A
a
and let
B
b
.
