and the Exponential Function
Disclaimer: these notes are not mathematically rigorous.
Instead, they present quick, and, I hope,
plausible, derivations of the properties of e, e
and the natural logarithm.
Consider the following series:
runs through the positive
integers. What happens as
gets very large?
It’s easy to find out if you use a scientific calculator having the function x^y.
The first three terms are 2,
You can use your calculator to confirm that for
= 10, 100, 1000, 10,000, 100,000,
1,000,000 the values of
are (rounding off) 2.59, 2.70, 2.717, 2.718, 2.71827, 2.718280.
calculations strongly suggest that as
goes up to infinity,
goes to a definite limit.
It can be
proved mathematically that
go to a limit, and this limiting value is called
The value of
is 2.7182818283… .
To try to get a bit more insight into
, let us expand it using the binomial theorem.
Recall that the binomial theorem gives all the terms in (1 +
, as follows:
To use this result to find
, we obviously need to put
We are particularly interested in what happens to this series when
gets very large, because that’s when
we are approaching
In that limit,
tends to 1, and so does
, we can ignore the
-dependence of these early terms in the series altogether!
When we do that, the series becomes just:
And, the larger we take
, the more accurately the terms in the binomial series can be simplified in this
way, so as