Three equivalent denitions of e
Gilles Cazelais
There are several ways to dene the number e. Three common denitions are the following.
1. e =
k=0
1
k!
= 1 +
1
1!
+
1
2!
+
1
3!
+
2. e = lim
n
_
1 +
1
n
_
n
3. e is the positive real number such that
_
e
1
1
t
dt = 1.
We will show that all three denitions are well-dened
and equivalent.
Lets rst consider the sequence
s
n
=
n
k=0
1
k!
= 1 +
1
1!
+
1
2!
+
1
3!
+ +
1
n!
.
It is clear that {s
n
} is an increasing sequence, i.e., s
n
< s
n+1
for all n = 0, 1, 2, . . . Starting from n = 1, we
have n! = 1 2 3 n 1 2 2 2 = 2
n1
, therefore
s
n
1 + 1 +
1
2
+
1
2
2
+ +
1
2
n1
< 1 +
k=0
1
2
k
= 1 +
1
1
1
2
= 3.
Since {s
n
} is increasing and bounded above, then lim
n
s
n
exists.
Now lets consider the sequence
t
n
=
_
1 +
1
n
_
n
.
From the binomial theorem, we have
t
n
=
n
k=0
_
n
k
__
1
n
_
k
= 1 + n
_
1
n
_
+
n(n 1)
2!
_
1
n
2
_
+
n(n 1)(n 2)
3!
_
1
n
3
_
+ +
n(n 1)(n 2) 1
n!
_
1
n
n
_
= 1 + 1 +
1
2!
_
1
1
n
_
+
1
3!
_
1
1
n
__
1
2
n
_
+ +
1
n!
_
1
1
n
__
1
2
n
_
_
1
n 1
n
_
.
Observe that for all k = 1, 2, . . . , n 1, we have
0 <
_
1
k
n
_
< 1, and
_
1
k
n
_
<
_
1
k
n + 1
_
.
The term well-dened is used for something that is dened in an unambiguous way. For example, if we dene a certain
number as the limit of a sequence, the denition is well-dened provided that we can show that the limit exists.
1
It is then clear that for all n = 1, 2, 3, . . . we have
t
n
< t
n+1
and t
n
s
n
< 3.
Since {t
n
} is increasing and bounded above, then lim
n
t
n
exists and satises
lim
n
t
n
lim
n
s
n
. (1)
Now let m be a xed integer such that 2 m n, then
1 + 1 +
1
2!
_
1
1
n
_
+ +
1
m!
_
1
1
n
__
1
2
n
_
_
1
m1
n
_
t
n
.
By taking the limit as n we deduce that
s
m
= 1 +
1
1!
+
1
2!
+
1
3!
+ +
1
m!
lim
n
t
n
.
By taking the limit as m we get
lim
m
s
m
lim
n
t
n
. (2)
Combining inequalities (1) and (2), we deduce that
lim
n
s
n
= lim
n
t
n
.
This establishes the equivalence of denitions 1 and 2.
Now lets consider the function
f(x) =
_
x
1
1
t
dt, for x > 0.
Since 1/t is positive and continuous over t > 0, then f(x) is a strictly increasing dierentiable function
satisfying
f
(x) =
1
x
.
Since f(1) = 0 and f(4)
1
2
+
1
3
+
1
4
=
13
12
> 1, we see that denition 3 is well-dened since we can guarantee
the existence of a unique number 1 < e < 4 such that
_
e
1
1
t
dt = 1.
0
t
y
1
1 2 3 4
y = 1/t
The function f(x) has the property that for any n and any x > 0, we have
f(x
n
) = nf(x). (3)
Proof. Let g(x) = f(x
n
) and h(x) = nf(x). Since f(1) = 0, we have g(1) = h(1) = 0. From the chain rule
we deduce that
g
(x) = f
(x
n
) nx
n1
=
1
x
n
nx
n1
=
n
x
and
h
(x) = nf
(x) =
n
x
.
Since g
(x) = h
(x) for all x > 0 and g(1) = h(1), then g(x) = h(x) for all x > 0.
2
Let
a = lim
n
_
1 +
1
n
_
n
.
Since f is continuous, then
f(a) = f
_
lim
n
_
1 +
1
n
_
n
_
= lim
n
f
__
1 +
1
n
_
n
_
.
From property (3), we get
f(a) = lim
n
nf
_
1 +
1
n
_
= lim
n
f
_
1 +
1
n
_
1
n
= lim
n
f
_
1 +
1
n
_
f(1)
1
n
(since f(1) = 0)
= lim
h0
f (1 + h) f(1)
h
= f
(1)
f(a) = 1.
This establishes the equivalence of denitions 2 and 3. Therefore, all three denitions of e are equivalent.
Observe that denition 2 is equivalent to
lim
h0
+
(1 + h)
1/h
= e.
To study lim
h0
(1 + h)
1/h
, observe that
lim
h0
(1 + h)
1/h
= lim
k0
+
(1 k)
1/k
= lim
n
_
1
1
n
_
n
.
Now,
_
1
1
n
_
n
=
_
n 1
n
_
n
=
_
n
n 1
_
n
=
_
1 +
1
n 1
_
n
=
_
1 +
1
n 1
_
n1
_
1 +
1
n 1
_
.
As n , we have
_
1 +
1
n1
_
n1
e and
_
1 +
1
n1
_
1. Therefore,
lim
n
_
1
1
n
_
n
= e.
3
We can conclude that
lim
h0
(1 + h)
1/h
= e.
This is equivalent to the important limit
lim
h0
e
h
1
h
= 1.
We can use this limit to prove that
d
dx
e
x
= e
x
.
Proof.
d
dx
e
x
= lim
h0
e
x+h
e
x
h
= lim
h0
(e
h
1)e
x
h
=
_
lim
h0
e
h
1
h
_
e
x
= 1 e
x
= e
x
.
Now let ln x = log
e
x so that
e
ln x
= x, for all x > 0.
Using the chain rule we deduce that
e
ln x
d
dx
ln x = 1 =
d
dx
ln x =
1
x
.
We can use the formula in denition 1 to numerically estimate e. For example by computing
1 +
1
1!
+
1
2!
+ +
1
100!
we obtain
e 2.718 281 828 459 045.
To learn more about the number e from a historical point of view, I recommend the wonderful book [1].
References
[1] Eli Maor, e: The Story of a Number, Princeton University Press, (1994)
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