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In-Depth Information
28
f
(
x
)
f
(
x
)
1. Convergence not uniform.
29 Let
S
bean uppr sum for
f
. Then
S
(
b
a
) is an upper sum for
f
.
Let
s
(
b
a
) is a lower sum for
f
.Itis
possibleto take
n
suMciently large so that
bea lowr sum for
f
. Then
s
/4(
b
a
). Since
f
is
s
integrable there are upper and lower sums such that
S
from
10.23. Then (
S
(
b
a
))
(
s
(
b
a
))
.
So
f
is integrable by qn 10.24.
30 The argument in qn 24 does not appeal to continuity.
31 Let (
f
)
f
, then by qn 30,
lim
F
(
x
)
lim
f
f
F
(
x
) (say).
F
(
x
)
F
(
x
)
(
f
f
) and, by qn 10.35,
F
(
x
)
F
(
x
)
f
f
.
If
n
N
sup
f
(
x
)
f
(
x
)
:
a
x
b
/(
b
a
), then
n
N
F
(
x
)
F
(
x
)
.
32 Thepointwiselimit function is given by
f
(
x
)
0.
f
(0)
1,
f
(0)
0. Yes.
33
f
(
x
)
0.
f
(
x
)
f
(
x
)
1/
n
, so convergence is uniform.
f
(
x
)
sin
nx
which is not pointwise convergent for any
x
0.
f
(sin
nx
)/
n
. Limit of integral
integral of limit by qn 30.
34 (i) Question 20, (ii) qn 10.39, (iii) qn 24, (iv) qn 10.51, (v) qn 3.54(v), the
difference rule, (vi) Fundamental Theorem of Calculus, qn 10.54, (vii) qns
24 and 31.
35
(i) Question 5.86.
(ii)
e
(
x
)
e
(
x
)
x
/(
n
1)!
...
x
/(
n
m
)!
x
/(
n
1)!
...
x
/(
n
m
)!
a
/(
n
1)!
...
a
/(
n
m
)!
e
(
a
)
e
(
a
).
(iii) Question 3.76, the inequality rule.
(iv) (
e
(
a
)
e
(
a
))
0as
n
.
(vi) let
f
e
, then
f
e
.(
f
)
e
uniformly and (
f
)
e
uniformly.
From qn 34(vi),
e
e
.
36
u
is absolutely convergent by (i), (ii) and the first comparison test, qn
5.26. So by qn 5.67 (absolute convergence) the series is convergent for all
