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(ii) If the Taylor series is convergent and the sequence of remainders
tends to 0 as
n
tends to infinity, then the Taylor series converges to
the function as we have seen in questions 38
—
41. In fact, when the
remainders tend to 0, the Taylor series is necessarily convergent.
(iii) Somewhat surprisingly, a Taylor series may be convergent
without
the remainders tending to zero. In this case, the Taylor series
cannot converge to the function. There is an example of this in qn
44.
43 Write down the Maclaurin series for the function
f
(
x
)
(1
x
).
Does it coincide with the binomial expansion of qn 5.98 for
a
?
If the Maclaurin series is
f
(
x
)
1
a
x
a
x
a
x
. . ., check
that
a
(
n
)/(
n
1),
so that the absolute values of the terms are monotonic decreasing.
Now check that
a
is an alternating series and that
a
/
a
·
·
·
· ...·
·
, so that
a
1/2
n
) is a null sequence. Use the alternating series test to
provethat the Maclaurin series for
f
(1) is convergent. How many
terms of this series are needed before
2 has been found correct to
one place of decimals, that is, until the error is less than 0.05?
44 (
Cauchy
, 1823) Thefunction
f
: R
R defined by
and (
a
when
x
0,
when
x
0,
has thecurious proprty that
f
(0)
0 for every value of
n
.
[A proof is given in Scott and Tims, p. 335 and another in
Bressoud pages 90 and 91.]
What is the Maclaurin series for this function?
Is there a non-zero value of
x
for which the remainders can form a
null sequence?
Construct an infinity of different functions each of which has the
same Maclaurin series.
A Taylor series is a valid description of the function from which it
originates if and only if its sequence of remainders is null. It is so
important to be able to prove that the sequence of remainders of a
Taylor series is null that a number of different forms of the remainder
have been devised since the Lagrange form of the remainder, which we
have been using, is not always suMcient to decide the matter, as we
found in qn 41.
(45) With thenotation and conditions of qn 35 lt
e
0
f
(
x
)
(
b
a
)
f
(
b
)
f
(
a
)
(
b
a
)
f
(
a
)
...
1)!
f
(
a
)
(
n
K
.
b
a
