### Motivation

My previous post about the definition of the exponential function has
provided *no* connection between a well-known characterization (or an
alternative definition) of the exponential function:

The term to be summed is simpler than the one in the binomial expansion of $(1 + s / n)^n$.

### Solution

We want this sum to be as small as possible as $n \to \infty$.

Observe that the fraction is a product

The reminds me a first-order approximation inequality

for each $x_k \in [0,1], k = 1,\dots,n$.

I call this “first-order approximation” because if you put $x_k = c_k \varepsilon$ for each $k = 1,\dots,n$, the LHS would become the first-order approximation of the product on the RHS.

This inequality is a nice induction exercise. The base case for $n = 1$ is trivial. The inductive step is a bit tricky. We can’t apply the case for $n = 2$ first because we aren’t sure what the range of $x_1 + \dots + x_n$ is. Instead we apply the induction hypothesis on $1 - (x_1 + \dots + x_n)$.

Put $x_i = \frac{i}{n}$ for each $i = 1,\dots,k$

Put this back into the sum that we want to minimize.

In the last line, we used the fact that the series on the right is absolutely convergent, which I have already shown in equations (4), (5) and (10) in my previous post. To end this post, take $n \to \infty$.