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Give reasons for each of the following propositions.
(i) Thefunction
is continuous.
(ii) Each function
f
and thefunction
areintegrableon [
a
,
b
].
(iii)
lim
f
.
(iv)
lim
(
f
(
x
)
f
(
a
))
(v)
f
(
x
)
f
(
a
).
(vi)
f
(vii)
(
f
)
f
uniformly on [
a
,
b
].
The conditions of qn 34 appear to be quite restrictive. But we need
to apply it, typically, to prove theorems about power series and, in this
case, when the functions concerned are polynomials, someof the
conditions we have used are obvious. One of the issues at stake is when
we may integrate or differentiate a power series term by term while still
guaranteeing its continuity or differentiability. Our theorems relating
integration and differentiation to uniform convergence will enable us to
decide this matter.
Uniform convergence of power series
A series of functions is said to be uniformly convergent when its
sequence of partial sums is uniformly convergent.
35 Let
e
(
x
)
1
x
x
/2!
...
x
/
n
!.
(i) Why is the sequence (
e
(
x
)) convergent for each
x
?
Wedefine
e
(
x
)
lim
e
(
x
).
a
x
a
, then
x
a
If
/
n
!
/
n
!.
(ii) Provethat
e
(
x
)
e
(
x
)
e
(
a
)
e
(
a
)
(iii) Let
m
in (ii) and provethat
e
(
x
)
e
(
x
)
e
(
a
)
e
(
a
) for all
x
in [
a
,
a
].
(iv) Provethat (
e
a
,
a
].
(v) Useqn 20 to provethat thefunction
e
is continuous on
[
a
,
a
].
(vi) Useqn 34 to provethat
e
(
x
)
e
(
x
).
)
e
uniformly on [
The earlier results of this chapter lead to the proof of the uniform
convergence of the sequence (
e
) in qn 35 on thebasis of theinequality
x
/
n
!
a
/
n
! and the convergence of
a
/
n
!. Weisolatethis claim in
the basic theorem about the uniform convergence of series.
