Exercise:Prove that is irrational.
Definition:We define an object called by
This is an example of an object called a power series. ``Power'' because
it is made of powers of and ``series'' for obvious reasons. For
us, a power series is not much different from a polynomial - we just
have infinetely many terms (pieces) at the same time. We still can
multiply and add power series according to the same rules.
Exercise:Check that
, so we can write
Exercise:Prove that .
Consider now the ring of polynomials in two variables and
. These are objects of the type
Exercise:Work out the addition, the multiplication, the
differentiations with repect to and
.
The main property of the exponential function is
Exercise:Work out the details.
Our next goal is to compute . Namely, if
is a polynomial,
what would be
?
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Remark:In calculus, the formula
is called the Taylor
expansion formula. It is an important formula, and it includes many
others as special cases. For example, if
is a polynomial then
as we saw this is just the Newton binomial formula.
If
then the Taylor expansion formula (where
is set
to zero) becomes
4
(with
replaced by
) - just
the formula for the geometric progression.
Note that in calculus, the Taylor expansion formula does not necessarily
hold for all values of . But in algebraic formal sense it always true.
Exercise:a) For what values of is the formula
4
true?
b) Let for
and
for
. Use
calculus to show that
for all
.
Now let us consider two operation: and
.
Exercise:Let and
are two letters such that
. Prove that
by multiplying the corresponding series and moving
all
's to the right and all
's to the left.
Remark:Our notations are not completely random - is the standard notation
for the Plank constant, and
is traditionally related to the word
quantum.