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:
$$\exp(s) = \lim_{n\to\infty} \sum_{k=1}^n \frac{s^k}{k!}.$$
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 $n \to \infty$.
$$\sum_{k=2}^n \left( 1 - \frac{n \cdot \dots \cdot (n + 1 - k)}
{\underbrace{n \cdot \dots \cdot n}_{n^k}} \right) \, \frac{x^k}{k!}$$
Observe that the fraction is a product
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