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.