A previous web page defined
two particular sequences converging to a number -
which by experiment can be shown to be approximately 2.718281828.
The very reasonable question arises asking if this number really is
the Euler number
for all
An alternative approach is to define the limit of the infinite series
1+and prove that this limit is precisely
For any doubting students of analysis, or indeed just for
interested readers, this web page explores these ideas further. Just
for fun, I have chosen to take a very different approach to either of
the more normal ones just outlined and will define and use the natural
logarithm function, and apply that to the function
We start by defining two functions using a limiting process similar to things you have already seen.
Suppose
and
We will spend a bit of time proving that both these limits exist and in fact they are equal.
To this end, we fix some positive
Lemma.
For all
Proof.
Indeed, we have
Lemma.
For all sufficiently large
Proof.
Let
by the exponential inequality, hence
as required.
Proof.
Let
by the exponential inequality, hence
and so
as required.
Proposition.
For all real
Proof.
The monotone convergence theorem together with the three
preceding lemmas already show that the limits
Therefore by continuty of division at (,) and
uniqueness of limits
Definition.
For
Having defined this function, we want to explore some of its properties.
In particular, we would like to know that it behaves as a logarithm function
and is the inverse to the
Lemma.
For each
Proof.
using the continuity of scalar multiplication.
Lemma.
For each
Proof.
We have
and this together with
Proposition.
The function
Proof.
Suppose
The argument just given does not show that
Proposition.
For each
Proof.
By definition
for eventually all
and
where both sequences _{
} converge
to
Proposition.
The function
Proof.
We already know
The case when
Continuity follows from a general result that says
a nondecreasing function
The notion of derivative is outside the scope of these notes. However, you may be familiar enough with it to understand the following argument.
Proposition.
The function
Proof.
By definition,
or, as
which equals
as required.
Proposition.
The function