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Lecture 43:
Multiplication and Division of Power Series
Dan Sloughter
Furman University
Mathematics 39
May 20, 2004
43.1
Multiplication of power series
The following generalization of the power rule is known as
Leibniz’s rule
.
Theorem 43.1.
If
f
and
g
are
n
times diﬀerentiable at
z
, then
d
n
dz
n
f
(
z
)
g
(
z
) =
n
X
k
=0
±
n
k
²
f
(
k
)
(
z
)
g
(
n

k
)
(
z
)
.
Proof.
When
n
= 1, the result is just the product rule:
d
dz
f
(
z
)
g
(
z
) =
f
(
z
)
g
0
(
z
) +
f
0
(
z
)
g
(
z
)
.
Assuming the result is true for
n
≥
1, we have
d
n
+1
dz
n
+1
f
(
z
)
g
(
z
) =
d
dz
n
X
k
=0
±
n
k
²
f
(
k
)
(
z
)
g
(
n

k
)
(
z
)
=
n
X
k
=0
±
n
k
²
(
f
(
k
)
(
z
)
g
(
n

k
+1)
(
z
) +
f
(
k
+1)
(
z
)
g
(
n

k
)
(
z
)
)
=
f
(
z
)
g
(
n
+1)
(
z
) +
n
X
k
=1
±±
n
k
²
+
±
n
k

1
²²
f
(
k
)
(
z
)
g
n

k
+1
(
z
)
+
f
(
n
+1)
(
z
)
g
(
z
)
1
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View Full Document =
n
+1
X
k
=0
±
n
+ 1
k
²
f
(
k
)
(
z
)
g
n

k
+1
(
z
)
,
which is the result for the (
n
+ 1)st derivative.
Now suppose
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This note was uploaded on 11/23/2009 for the course MATHEMATIC Mathematic taught by Professor Dansloughter during the Spring '04 term at Furman.
 Spring '04
 DanSloughter
 Math, Division, Multiplication, Power Rule, Power Series

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