Graphics Reference
In-Depth Information
44 The Maclaurin series is 0
....
The remainder always equals the whole value of the function, so
remainders are constant and not null.
For each choice of the constant
k
, thefunction
x
0
0
sin
x
k
·
f
(
x
) has the
same Maclaurin series as sin
x
.
45 As in qn 35,
F
(
a
)
F
(
b
)
0, and
F
continuous on [
a
,
b
] and differentiable
on (
a
,
b
),
(
b
c
)
(
n
1)!
f
(
c
)
K
0.
F
(
c
)
0
46 Using the work of qn 41, Cauchy's form of the remainder
(
x
x
)
x
(
n
1!)
R
(
x
)
(
1)
(
n
1)!
(1
x
)
1
·
1
(
1)
x
x
.
1
x
1
Now
1
x
x
0
1
1
x
(1
)/(1
x
)
1.
So
1
x
R
(
x
)
x
/(1
x
)
x
/(1
x
)
. But, if
x
1, this
last expression gives a null sequence, so (
R
(
x
)) is a null sequence.
