1. The Euler number e
This section is devoted to two beautiful monotonic sequences
_{
=
}
1+1
^{
}
and
_{
=
}
1+1
^{
+1
}
and their limits. Staring at the sequences, it is not clear
what will happen: will the 1+1
go to 1 so fast that
the exponent of or +1 not matter? or will
the exponent beat the 1+1
and the sequence go to infinity?
In fact, the answer is something in between. These sequences come
up regularly enough that it is worth learning the sequences and the
limits, if not the following rather pretty argument.
1+1
^{
}
1+
1
=1+1
1+1
^{
}
1+1
^{}
showing that (_{
)
} is monotonic nondecreasing.
Now let 1
again and consder
a second application of the exponential inequality:
11+1
+1
+1
1
+1
+1
1+1
=1
1
+1
+1
^{
+1
}
+1
^{
+1
}
1+1
^{
+1
}
1+1
^{
+1
}.
It follows that (_{
)
} is monotonic nonincreasing.
Also _{
=
1+1
}
for all so given any ,
and
,
we have
_{
}_{
}
Hence _{
} for all ,
and therefore
both sequences (_{
)
} and (_{
)
} converge,
to _{
1
} and _{
2
}, say,
where
_{1
2
}
. But
_{

}=
1+1
^{
1+11
}=
_{
}
so _{
=
+
1
1+
10=
1
}
by continuity, so
_{1=
2
}
by uniqueness of limits.
The limit ( =
_{1=
2
}
) that is the limit here
can be calculated approximately using the terms of the sequences (though there
are better methods that give more accurate answers more quickly) and it turns out
that =2.718281828..., the number commonly (pun intended!) used
as the base of the natural log.
Theorem.
The sequences _{
=
}
1+1
^{
}
and
_{
=
}
1+1
^{
+1
} both converge to =2.718281828....
Taking this idea a little further it is possible to define
a function ()=
1+
^{
}. It turns out that ()
has all the properties expected
of an exponential function, including (0)=1,
(1)=
and (+)=()
()
and
we may define
^{
=()
}
. Once the
inverse function or of
()
has been defined we may define
^{
=
()
}
for
0
and arbitrary
. But apart
from stating the essential ideas here, this takes us far to far
for this module, so I shall omit the details.
Exercise.
Modify the arguments at the top of this web page
to show that for each
the sequence
1+
^{
}
is eventually monotonic nondecreasing and the sequence
1+
^{
+1
}
is eventually monotonic nonincreasing.