An extraordinary limit

Limits part 1: Infinite sequences: Standard limits

An extraordinary limit

In this section we will study the sequence $a_n = \left( 1 + \tfrac{1}{n}\right)^n$ . This animation suggests that it converges and that its limit is approximately equal to $2.7$ .

For any $x$ we have:

$\lim_{n\to\infty} \left( 1 + \frac{x}{n} \right)^n = e^x$

To prove this, we actually first have to consider how the number $e$ is defined. It is even common to agree that $e$ is the number that is the limit of this sequence. This theorem then changes into a mathematical definition.

Another option is to first define the natural logarithm $\ln(x)$ as a primitive of $f(x) = \frac{1}{x}$ and then set $e$ to be the unique solution of $\ln(x) = 1$ . The idea of the proof in this case is that we squeeze the sequence:

$\frac{e}{1+\frac{1}{n}} \leq \left( 1 + \frac{1}{n}\right)^n \leq e$

Substituting $x=1$ we obtain

$\lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n = e$

The number $e$ is a mathematical constant that approximately equals $2.71828$ . The number $e$ has the special property that the function $f(x) = e^x$ is its own derivative.