Links forward

Series for \(e\)

One way of defining the number \(e\) is

\[ \lim_{n \to \infty} \Big(1+\dfrac{1}{n}\Big)^n. \]

The binomial theorem gives

\[ \Big(1+\dfrac{1}{n}\Big)^n = \dbinom{n}{0}+ \dbinom{n}{1}\dfrac{1}{n} + \dbinom{n}{2}\Big(\dfrac{1}{n}\Big)^2 + \dots + \dbinom{n}{r}\Big(\dfrac{1}{n}\Big)^r + \dots + \dbinom{n}{n-1}\Big(\dfrac{1}{n}\Big)^{n-1} + \dbinom{n}{n}\Big(\dfrac{1}{n}\Big)^n. \]

Using the result that

\[ \lim_{n \to \infty} \dbinom{n}{k}\Big(\dfrac{1}{n}\Big)^k = \dfrac{1}{k!}, \]

we obtain the result

\[ e = \lim_{n \to \infty} \Big(1+\dfrac{1}{n}\Big)^n = \dfrac{1}{0!} + \dfrac{1}{1!}+ \dfrac{1}{2!}+\dfrac{1}{3!} + \dotsb. \]

Formally, this argument involves some difficult limiting processes.

Next page - Links forward - The generalised binomial theorem