SECTION 11.7: POWER SERIES
A series of the type
(1)
∞
n
=0
a
n
x
n
is called a power series. Here
a
n
are fixed real numbers, while
x
is a variable.
a
n
is
the coefficient of
x
n
.
The nature of the series (1) is clearly influenced by the particular value of
x
.
The set of those
x
for which the power series converges is called the
domain of
convergence
. We denote it by
S
. It is also called the
interval of convergence
(since
it turns out to be an interval).
Example:
the power series
∞
n
=0
x
n
= 1 +
x
+
x
2
+
x
3
+
. . .
is the well known geometric series which converges only for

1
< x <
1.
In other words, in this particular case,
S
= (

1
,
1).
We also know the sum of this series when
x
∈
S
, namely
1
x

1
. Example:
∑
∞
n
=0
(

1
2
)
n
=
1
1+1
/
2
=
3
2
(
x
=

1
2
)
∑
∞
n
=0
2
n
is divergent
(
x
= 2).
Theorem 1
(Fundamental Theorem J. Hadamard)
.
Given a power series
∞
n
=0
a
n
x
n
,
there exists a number
R
(radius of convergence) such that the power series is:
•
Absolutely convergent for

R < x < R
•
Divergent for

x

> R
•
The endpoints
x
=

R
and
x
=
R
need further investigation.
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 Spring '11
 Prof.Ramachandran
 Power Series, Mathematical Series, lim, convergent, Theorem J. Hadamard

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