State and prove tauber theorem for uniform eonvergence
Answers
Answer:
proof of Tauber’s convergence theorem
Let
f
(
z
)
=
∞
∑
n
=
0
a
n
z
n
,
be a complex power series, convergent in the open disk
|
z
|
<
1
. We suppose that
1.
n
a
n
→
0
as
n
→
∞
, and that
2.
f
(
r
)
converges to some finite
L
as
r
→
1
−
;
and wish to show that
∑
n
a
n
converges to the same
L
as well.
Let
s
n
=
a
0
+
⋯
+
a
n
, where
n
=
0
,
1
,
…
, denote the partial sums of the series in question. The enabling idea in Tauber’s convergence result (as well as other Tauberian theorems) is the existence of a correspondence in the evolution of the
s
n
as
n
→
∞
, and the evolution of
f
(
r
)
as
r
→
1
−
. Indeed we shall show that
∣
∣
∣
s
n
−
f
(
n
−
1
n
)
∣
∣
∣
→
0
as
n
→
∞
.
(1)
The desired result then follows in an obvious fashion.
For every real
0
<
r
<
1
we have
s
n
=
f
(
r
)
+
n
∑
k
=
0
a
k
(
1
−
r
k
)
−
∞
∑
k
=
n
+
1
a
k
r
k
.
Setting
ϵ
n
=
sup
k
>
n
|
k
a
k
|
,
and noting that
1
−
r
k
=
(
1
−
r
)
(
1
+
r
+
⋯
+
r
k
−
1
)
<
k
(
1
−
r
)
,
we have that
|
s
n
−
f
(
r
)
|
≤
(
1
−
r
)
n
∑
k
=
0
k
a
k
+
ϵ
n
n
∞
∑
k
=
n
+
1
r
k
.
Setting
r
=
1
−
1
/
n
in the above inequality we get
|
s
n
−
f
(
1
−
1
/
n
)
|
≤
μ
n
+
ϵ
n
(
1
−
1
/
n
)
n
+
1
,
where
μ
n
=
1
n
n
∑
k
=
0
|
k
a
k
|
are the Cesàro means of the sequence
|
k
a
k
|
,
k
=
0
,
1
,
…
Since the latter sequence converges to zero, so do the means
μ
n
, and the suprema
ϵ
n
. Finally, proved