Graphics Reference
In-Depth Information
36
The Weierstrass M
-
test
Welt
f
(
x
)
u
(
x
) for
x
A
R, and assume
(i) for each
n
, and for all
x
A
, there exists
M
such that
u
(
x
)
M
, and
(ii)
M
is convergent.
We investigate the convergence of the sequence (
f
).
Explain why the series
u
(
x
) is absolutely convergent for all
x
A
.
Deduce that the sequence (
f
) is pointwise convergent to a function
f
, say.
Provethat
u
(
x
)
M
,
and deduce that
(
x
)
f
(
x
)
f
M
M
,
so that, letting
m
,
(
x
)
f
(
x
)
f
M
M
.
Now deduce that (
f
) converges uniformly.
37 Use the Weierstrass
M
-test to prove that each of the following
series is uniformly convergent on the domain given.
(i)
x
/
n
on thedomain [
1, 1],
(ii)
,
(iii)
1/(
n
x
) on thedomain R,
(iv)
(sin
nx
)/2
on thedomain
R
a
,
a
],
(v)
(
x
n
)/(
x
n
) on thedomain [
a
,
a
],
(vi)
x
/(
x
n
) on thedomain [
(
n
1)
x
on thedomain [
a
,
a
], where 0
a
1,
(vii)
x
(1
x
)/
n
on thedomain [0, 1].
38 Let
f
.
For
1
x
1, find thepointwiselimit function
f
of the sequence
(
f
(
x
)
1
x
x
...
x
).
For
x
in this range, prove that
f
(
x
)
f
(
x
)
x
/(1
x
).
Deduce that the convergence of the sequence (
f
) is not uniform on
thedomain (
1, 1).
Prove that the convergence of the sequence (
f
) is uniform on the
domain [
a
,
a
], provided 0
a
1.
