Graphics Reference
In-Depth Information
a
/
n
x
x
x
82
. Convergent when
1 and divergent when
x
1 by Cauchy's
n
th root test. Convergent when
x
1 by the
alternating series test. Divergent when
x
1, harmonic series.
/(
83
a
x
n
)
x
. Conv./div. as
x
/
1. Absolutely
convergent when
x
1.
84 Conv./div. as
1 by Cauchy's
n
th root test. Convergent by
alternating series test when
x
/
x
1.
a
x
n
x
x
1. When
x
85
. Conv./div. as
/
1,
1 from qn 32. When
x
convergent when
1, convergent when
0 by alternating series test.
86
a
/
a
x
/(
n
1)
0 for all values of
x
.
87
a
/
a
x
/(2
n
1)(2
n
)
0 for all values of
x
.
a
x
(
n
unless
x
88
/
a
1)
0.
89
a
2
x
. Con./div. as
x
/
. Divergent when
x
since
terms not null.
a
2
x
/
n
2
x
. Conv./div. as
x
/
90
. Convergent when
x
by the alternating series test. Divergent when
x
, harmonic
series.
91
a
x
x
/
y
·
a
y
and
x
/
y
1.
92
a
x
x
/
y
·
a
y
K
·
x
/
y
.
93 If
a
x
were convergent with
x
y
, then by qn 91,
a
y
would
be convergent. Contradiction.
94
C
unbounded
for any given
y
, there is an
x
with
x
y
and
a
x
convergent, so
a
y
must be convergent for all
y
by qn 91.
If
C
is bounded, let
R
x
, then by qn 4.64, there is a
y
with
a
y
convergent with
R
y
R
, and then
a
x
is absolutely
convergent by qn 91.
x
C
x
R
.
95
a
·
x
kx
. Convergent when
kx
1,
divergent when
kx
1, by d'Alembert's ratio test.
x
/
a
x
a
/
a
(
96
a
x
a
) ·
x
kx
.
