On Exponential Function

18 Dec, 2023

One reason I avoid to introduce the Euler constant by the famous Bernoulli limit

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

is that despite its brievity, to prove its convergence in the class can be somewhat cumbersome. Normally I would introduce exponential function as the sum of the Taylor serie:

e^{x} = \sum_{k = 0}^{\infty} \frac{1}{k!} x^{k} .

The only two key points I would prove in the class would be that

By ratio test or whatsoever, the serie converges: actually $\frac{a_{k + 1}}{a_{k}} = \frac{x}{k + 1}$ , for any fixed $x$ , the ratio goes to zero.
Denote the polynomial by $f (x)$ , we automatically have $f (x) f (y) = f (x + y)$ . This is obvious as we collect the term $x^{m} y^{n}$ , its coefficient would be $\frac{1}{n!} \frac{1}{m!} = \frac{1}{(m + n)!} (\binom{n + m}{n})$ , now apply binomial theorem, we would reach to the result easily.

Thus, $f$ is a legitimate exponential function, and we define $e = f (1)$ . A nice aspect of this is that we have $(e^{x})^{'} = e^{x}$ for free, if we don't bother with the Fubini theorem.

#"hs-maths"