Exercise 1.
Let
and
be integers
Consider
the power series
and put
a) By induction
on
show that
+...+
and, by
induction on
deduce the
expansion
where
denotes the
binomial coefficient
b) Using
show that
=
(This
generalises (1) in the case where
)
Exercise 2.
Determine the radius of convergence of each of the series:
a)
()
b)
( is a positive
integer)
c) with
for n≥0,
where a and b are real numbers such that
.
Exercise 3.
a) Prove the Unicity Theorem:
If
then
Exercise 4.
Prove the Uniqueness Theorem:
Suppose
and
are power series with the same radius
of convergence R.
If
= then
for n=0,1,2,…
Exercise 5.
Consider the power series
and
Put
Show the
following relations
and if
Exercise 6.
Let
C
,
not being an
integer
What is the
radius of
convergence of
Show that its
sum
for
satisfies the
differential equation
Exercise 7.
Let
be a power
series with
Put
for
and
Show that
i)
ii)
= for
Exercise 8.
Let
be a power
series with
for
where
and
are two given
real numbers.
a) Show that
for
we have
where
and deduce
that the radius of convergence
is nonzero.
b) Show that
for
and deduce
that for
we have
c)
Let
and
be two roots
of
Using the
decomposition of the right-hand side of (1), find an
expression of
using
and
and deduce
that
(Note that if
then
Exercise 9.
Prove the following inequality
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