Term by Term Operations on Series
If
f
(
x
)
=
∑
n
=
0
∞
a
n
x
n
has radius of convergence
M
∈
R
+
, then
f
is differentiable on
(
−
M
,
M
)
and the following are true:
∑
n
=
1
∞
n
a
n
x
n
−
1
has radius of convergence
M
such that for any
x
∈
(
−
M
,
M
)
we have
f
′
(
x
)
=
∑
n
=
1
∞
n
a
n
x
n
−
1
∑
n
=
0
∞
a
n
n
+
1
x
n
+
1
has radius of convergence
M
such that for any
x
∈
(
−
M
,
M
)
we have:
∫
0
x
f
(
t
)
d
t
=
∑
n
=
0
∞
a
n
n
+
1
x
n
+
1
Hadamard
Given a power series
∑
n
=
0
∞
a
n
x
n
then either there exists a set
C
such that the series converges for any
x
∈
C
and diverges otherwise.
C
is either of the form
[
−
M
,
M
]
or
R
itself, such that
For each
[
a
,
b
]
⊆
C
the series converges uniformly
If
α
=
lim sup
n
→
∞
|
a
n
|
1
n
, then
C
=
{
R
if
α
=
0
∅
if
α
=
+
∞
[
−
1
α
,
1
α
]
if
α
∈
R
+