Graphics Reference
In-Depth Information
0as
n
Provethat (
R
(
x
))
(i) when 0
x
1;
(ii) when
x
1;
(iii) when
x
0.
Deduce that
log(1
x
)
x
n
, when
x
1.
(
1)
In fact, the series expansion is valid for
1
x
1 as wewill
show in qns 46 and 11.38.
42 Use d'Alembert's ratio test (qn 5.69 or 5.95) to prove that
x
x
. . . is not convergent when
x
1? Deduce that, in
this case, the sequence (
R
x
(
x
)) in qn 41 is not a null sequence.
In qn 36,
h
2!
f
h
(
n
f
(
a
)
hf
(
a
)
(
a
)
...
1)!
f
(
a
)
is called the
polynomial expansion
of
f
(
a
h
) and
h
n
!
f
(
a
h
),
R
(
h
)
is called the
Lagrange form of the remainder
.
In qn 37,
x
2!
f
x
(
n
xf
1)!
f
f
(0)
(0)
(0)
...
(0)
is called the
polynomial expansion
of
f
(
x
) and
x
n
!
f
(
x
),
R
(
x
)
is called the
Lagrange form of the remainder
.
(i) If we regard the polynomial expansions as the partial sums of a
power series (known as the Taylor series in the case of qn 36 and
as the Maclaurin series in the case of qn 37) the power series may
be convergent or not. In any case Taylor's Theorem is still a valid
theorem and the remainder term measures how good an
approximation to the function is given by the polynomial. There is
an example of a non-convergent Taylor series in qn 42.
