The number \(e\)

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Section 6.3 The number \(e\)

The number \(e\text{,}\) together with \(0,1\) and \(\pi\text{,}\) is perhaps the most important of all the real numbers. It arose in practical calculations in compound interest when one wants to calculate the interest on a “continuous” basis rather than on a discrete (annually, monthly, daily, hourly) basis. It is in this context that it arises as the limit of the sequence \((a_n)\text{,}\) where

\begin{equation*} a_n = \left(1 + \frac{1}{n}\right)^n\text{.} \end{equation*}

In one of the exercises we shall establish that this sequence does indeed have a limit \(L\) such that \(2 \lt L \lt 4\text{.}\)

It was also used as the base for natural logarithms. These were invented as early as 1618 by John Napier in the Merchiston area of Edinburgh.

Finally, it arises as the infinite sum

\begin{equation*} e = \sum_{n=0}^\infty \frac{1}{n!}\text{,} \end{equation*}

which converges by the comparison test since for \(n \geq 1\)

\begin{equation*} \frac{1}{n!}=\frac{1}{n\cdot (n-1)\cdots 2\cdot 1}\leq \frac{1}{2\cdot 2\cdots 2\cdot 1 } = \frac{1}{2^{n-1}}\text{,} \end{equation*}

and the geometric series \(\sum_{n=1}^{\infty}\frac{1}{2^{n-1}}\) converges. This is the definition of \(e\) which we adopt officially in mathematics. In the exercises you will establish that

\begin{equation*} \sum_{n=0}^\infty \frac{1}{n!} = \lim_{n\to\infty} \left(1 + \frac{1}{n}\right)^n\text{,} \end{equation*}

showing that the two definitions of \(e\) which we have proposed are in fact the same.

What is not clear from either defintion of \(e\) is whether it is rational or irrational.