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The Number
e
and the Exponential Function
Michael Fowler,
UVa
Disclaimer: these notes are not mathematically rigorous.
Instead, they present quick, and, I hope,
plausible, derivations of the properties of e, e
x
and the natural logarithm.
The Limit
Consider the following series:
where
n
runs through the positive
integers. What happens as
n
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,
2.25, 2.37.
You can use your calculator to confirm that for
n
= 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.
These
calculations strongly suggest that as
n
goes up to infinity,
goes to a definite limit.
It can be
proved mathematically that
does
go to a limit, and this limiting value is called
e
.
The value of
e
is 2.7182818283… .
To try to get a bit more insight into
for large
n
, let us expand it using the binomial theorem.
Recall that the binomial theorem gives all the terms in (1 +
x
)
n
, as follows:
To use this result to find
, we obviously need to put
x
= 1/
n
, giving:
.
We are particularly interested in what happens to this series when
n
gets very large, because that’s when
we are approaching
e
.
In that limit,
tends to 1, and so does
.
So, for
large enough
n
, we can ignore the
n
-dependence of these early terms in the series altogether!
When we do that, the series becomes just:
And, the larger we take
n
, the more accurately the terms in the binomial series can be simplified in this
way, so as
n