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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
1
lim(1
)
n
n
n
e
→∞
+=
Consider the following series:
23
11
1
(1 1), (1+ ) , (1
) , .
.., (1
) ,.
..
n
n
++
+
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
1
(1
)
n
n
+
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,
1
)
n
n
+
goes to a definite limit.
It can be proved mathematically that
1
)
n
n
+
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
1
)
n
n
+
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:
)
)
(2
)
)
1
...
2!
3!
n
n
nn
n
x
nx
x
x
x
−
−−
+
=+ +
+
+ +
To use this result to find
1
)
n
n
+
, we obviously need to put
x
= 1/
n
, giving:
1
1
)
)
)
)
1
.
( )
...
3!
n
n
n
n
n
−
+
=+
+
+
+
.
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,
( )
2
1/
n
−
tends to 1, and so does
.
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 Fall '07
 MichaelFowler
 Taylor Series, Limit, Natural logarithm, Logarithm

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