Graphics Reference
In-Depth Information
Power series
Applications of d'Alembert's ratio test and Cauchy's nth root test for
absolute convergence
78 For what values of
x
is
x
convergent? divergent? See qn 9.
79 By using d'Alembert's test for absolute convergence (qns 68 and 69)
determine for what values of
x
the series
nx
is convergent and
for what values it is divergent.
80 By using Cauchy's
n
th root rest for absolute convergence (qn 70)
determine for what values of
x
the series
nx
is convergent and
for what values it is divergent.
n
x
81 For what values of
x
is the series
convergent and for what
values is it divergent?
82 For what values of
x
is the series
x
/
n
convergent and for what
values is it divergent? This question is particularly illuminating
because of the possibilities at the critical values when
x
1.
83 For what values of
x
is the series
x
/
n
convergent and for what
values is it divergent?
84 For what values of
x
is the series for arctan
x
convergent:
x
3
x
5
x
7
...
(
1)
x
2
n
1
...?
x
85 Discuss the convergence of the series
n
x
for various values of
x
and
.
x
86 Provethat
/
n
! is absolutely convergent for each value of
x
.
87 Provethat
(
1)
x
/(2
n
1)! is absolutely convergent for
each value of
x
.
88 Provethat
n
!
x
is only convergent when
x
0.
89 For what values of
x
is the series
(2
x
)
convergent and for what
values is it divergent?
90 For what values of
x
is the series
(2
x
)
/
n
convergent and for what
values is it divergent?
Radius of convergence
91 (
Abel
, 1827) If
a
y
is convergent, use qn 11 (the null sequence
test) to prove that, for suMciently large
n
,
a
y
1.
