Graphics Reference
In-Depth Information
1
{
}
1
n
1
n
=
lim sup
a
=
sup
a
(D.6a)
n
n
R
n
Æ•
or
1
a
a
n
+
1
=
lim
(D.6b)
R
n
Æ•
n
provided that the limits exist.
Proof.
See [Buck78].
Definition.
The number R in formulas (D.6) is called the
radius of convergence
of
the series (D.4).
Clearly, if R is the radius of convergence of the power series (D.4), then (D.5) will
converge for all x satisfying |x - c| < R, and diverge for all x satisfying |x - R| > R.
Therefore, the points at which a power series converges is an open interval together
with possibly its endpoints. The endpoints of the interval typically have to be checked
separately for convergence or divergence.
Definition.
The interval of numbers at which the power series (D.4) or (D.5) con-
verges is called its
interval of convergence
.
D.2.3. Theorem.
Let R be the radius of convergence of the series defined by (D.4).
The function it defines is differentiable for all x, |x| < R. Its derivative can be obtained
by termwise differentiation and its radius of convergence is again R.
Proof.
See [Buck78].
Let f : (a,b) Æ
R
be of class C
k
. Let x
0
Π(a,b). The polynomial
Definition.
1
k
()
()
+¢
()
-
(
)
+ ◊◊◊+
k
()
-
(
)
fx
f x
x x
f
xxx
0
0
0
0
0
k
!
is called the
Taylor polynomial of f of degree k at x
0
.
D.2.4. Theorem.
(The Taylor Polynomial Theorem) Let f : (a,b) Æ
R
be of class C
k+1
and let c Œ (a,b). Then for any x Œ (a,b) there is an aŒ[c,x] such that
1
1
k
k
+
1
()
=
()
+
()
-
(
)
+ ◊◊◊+
()
k
()
-
(
)
(
k
+
1
)
()
-
(
)
fx
fc
f c x c
fcxc
+
f
a
x
c
.
(
)
k
!
k
+
1
!
Proof.
See [Buck78].
Definition.
Let f : (a,b) Æ
R
be of class C
•
. Let P
c
(x) be the Taylor polynomial of f of
degree k at c Π(a,b). Let R
k
(x) = f(x) - P
c
(x). The function f is said to be
analytic
at c
if there is an open interval
I
in (a,b) containing c such that
