Graphics Reference
In-Depth Information
x
A
and
f
is well defined. The inequality with
m
terms is obtained by
repeated application of the triangle inequality (qn 2.61), the second
inequality is only a rewrite of the first, and the third inequality comes from
qn 3.76, the inequality rule. As
N
theright-hand sideof thelast
inequality tends to 0; see qn 5.17.
37 (i) Take
M
1/
n
, (ii)
M
(
)
, (iii)
M
1/
n
, (iv)
M
a
/
n
,
(v)
M
1/
n
a
/
n
, (vi)
M
(
n
1)
a
using Cauchy's
n
th root test, qn
1/2
n
5.35, or d'Alembert 5.43, (vii)
M
using differentiation and qn 2.45.
38
f
(
x
)
1/(1
x
). As
x
1
,
f
(
x
)
f
(
x
)
.
If
a
x
a
1,
x
a
M
.
39
(i) When
x
n
, there is a term in the series, after the
n
th, which is
greater than
. So although the series is convergent for each value of
x
by comparison with
1/
n
, the greatest difference between the
n
th
partial sum (i.e.
f
) and thelimit function is not null.
(ii) Likewise when
x
1/
(2
n
), there is a term in the series, after the
n
th, which is greater than 1. So although the series is convergent for
each value of
x
in (
1, 1), the greatest difference between the
n
th
partial sum and the limit function is greater than 1.
40 Thelimit function is continuous by qn 20.
41 A partial sum of a power series is a polynomial which is continuous. The
uniform convergence of the series on [0,
x
] gives the limit of the integral
equal to the integral of the limit from qn 24.
The radius of convergence of the two series is the same by qn 5.107.
42 Partial sums are polynomials, so derivatives of partial sums are also
polynomials. Both series have the same circle of convergence by qn 5.107
so, within that circle, both converge uniformly. This is the context in which
qn 34(vi) may beapplied.
43
(1
x
)
1
x
x
...
x
...
1
1
1
2
1
n
1
(
x
)
(
x
)
...
(
x
)
...
1
n
(
x
)
.
The radius of convergence
1.
m
n
Suppose(1
x
)
(
x
)
.
