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z
R
called the
circle of convergence
.
Qn 94 establishes that every power series has a radius of
convergence.
weobtain a circle
95 If (
a
)
k
0, prove that the radius of convergence of the
power series
/
a
a
x
is 1/
k
.
96 If (
a
)
k
0, prove that the radius of convergence of the
power series
a
x
is 1/
k
.
97 Find the radius of convergence of
(
nx
)
/
n
!.
98 (
Cauchy
, 1821) If
a
is a real number which is not a positive integer
or 0, and
a
n
a
(
a
1)...(
a
n
1)
n
!
, the binomial coeMcient,
a
n
prove that the power series
x
has radius of convergence 1.
99 If the sequence (
a
) is unbounded, show that, for any value of
x
0, infinitely many terms of the sequence are
1/
x
, so the
terms of the power series
a
x
do not form a null sequence.
Determine the radius of convergence.
100 If the sequence (
a
)
0as
n
, show that, for any valueof
x
0, and for suMciently large
n
, the terms of the sequence are
x
is absolutely
convergent for all values of
x
. Determine the radius of convergence
in this case.
/
x
, and hence show that the power series
a
101 Show that the series
((
x
)
) has radius of convergence
2. Notice that in this case, the sequence (
a
x
)
(
) is not convergent,
but oscillates between
and
.
Cauchy-Hadamard formula
Question 102 uses the notation of lim sup which was developed as
an optional part of thebook in qns 4.83 and 4.84.
102 If the sequence (
a
) is bounded but not necessarily convergent,
a
welt lim sup
A
. Wefurthr supposethat
A
0, sincethis
case has been considered in qn 99. Show that if
x
1/
A
then
a
1/
x
for infinitely many
n
and so the terms of the power
series
a
x
do not form a null sequence and the series is divergent.
Show also that if
x
1/
A
, then, choosing
r
such that
