This section is devoted to two beautiful monotonic sequences

and their limits. Staring at the sequences, it is not clear what will happen: will the $1+\frac{1}{n}$ go to 1 so fast that the exponent of $n$ or $n+1$ not matter? or will the exponent beat the $1+\frac{1}{n}$ 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.

We start by proving monotonicity. Let $1\u2a7dr\u2a7ds$ and apply the exponential inequality to $(1+\frac{1}{r}{)}^{r/s}$ giving

so

showing that *
$\left({a}_{n}\right)$ is monotonic nondecreasing*.

Now let $1\u2a7dr\u2a7ds$ again and consder a second application of the exponential inequality:

so

whence

It follows that *
$\left({b}_{n}\right)$ is monotonic nonincreasing*.

Also ${b}_{n}=\left(1+\frac{1}{n}\right){a}_{n}\u2a7e{a}_{n}$ for all $n$ so given any $k,l\in \mathbb{N}$ and $n\u2a7ek,l$ we have

Hence ${a}_{k}\u2a7d{b}_{l}$ for all $k,l$ and therefore both sequences $\left({a}_{n}\right)$ and $\left({b}_{n}\right)$ converge, to ${a}_{n}\to {e}_{1}$ and ${b}_{n}\to {e}_{2}$, say, where ${e}_{1}\u2a7d{e}_{2}\in \mathbb{R}$. But

so ${b}_{n}={a}_{n}+{a}_{n}\xb7\frac{1}{n}\to {e}_{1}+{e}_{1}\xb70={e}_{1}$ by continuity, so ${e}_{1}={e}_{2}$ by uniqueness of limits.

The limit $e$ ( $={e}_{1}={e}_{2}$ ) 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 $e=2.718281828...$, the number commonly (pun intended!) used as the base of the natural log.

Theorem.

The sequences ${a}_{n}={\left(1+\frac{1}{n}\right)}^{n}\text{and}{b}_{n}={\left(1+\frac{1}{n}\right)}^{n+1}$ both converge to $e=2.718281828...$.

Taking this idea a little further it is possible to define
a function $E\left(x\right)={lim}_{n\to \infty}{\left(1+\frac{x}{n}\right)}^{n}$. It turns out that $E\left(x\right)$ has all the properties expected
of an exponential function, including $E\left(0\right)=1$,
$E\left(1\right)=e$ and $E(x+y)=E\left(x\right)\xb7E\left(y\right)$ and
we may *define*
${e}^{x}=E\left(x\right)$. Once the
inverse function $log$ or $ln$ of
$E\left(x\right)$ has been defined we may define
${x}^{y}={e}^{ylog\left(x\right)}$ for
$x>0$ and arbitrary $y\in \mathbb{R}$. 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 $x\in \mathbb{R}$ the sequence ${\left(1+\frac{x}{n}\right)}^{n}$ is eventually monotonic nondecreasing and the sequence ${\left(1+\frac{x}{n}\right)}^{n+1}$ is eventually monotonic nonincreasing.