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39 Prove that each of the following series is not uniformly convergent
on thedomain given.
(i)
(
x
n
)/(
x
n
) on thedomain R,
(ii)
(
n
1)
x
on thedomain (
1, 1).
The heart of the proof lies in finding a value of
x
which shows that
the sum of the series from
n
to
cannot bemadearbitrarily small
by choiceof
n
, however large.
Compare these results with qn 37(v) and 37(vi).
It is in fact quite straightforward to show that any power series is
uniformly convergent within a closed interval inside its circle of
convergence. (
Abel
, 1827)
40 Suppose
R
is the radius of convergence of the power series
a
x
.
Show that if 0
a
R
then the series is uniformly convergent on
theintrval [
a
,
a
] by choosing a real number
r
such that
a
r
R
, and using the Weierstrass
M
-test with
M
r
.
Deduce that the limit function must be continuous on this domain.
a
41 Use qn 24 to show that if a power series is integrated term by term
then the integral of the limit function is equal to the limit of the
power series obtained by term-by-term integration, on an interval
[0,
x
] inside its circle of convergence.
Is the radius of convergence of the integrated series the same as
the radius of convergence of the original power series? See
qn 5.107.
42 If a power series is differentiated term by term show that the radius
of convergence of the differentiated series is the same as the radius
of convergence of the original series. See qn 5.107. Use qn 34 to
show that the limit function is differentiable and that the derivative
of the limit function is equal to the limit of the power series
obtained by term-by-term differentiation, within a closed interval
inside its circle of convergence.
43 By differentiating the series for 1/(1
x
)
m
times, establish the
binomial theorem for negative integral index.
44 By integrating the power series
1
x
x
...
x
...
