Euler's Number in Python

A comparative approach about the values provided by Python libraries for the Euler's number, in relation to the approximate value of e by a Taylor Polynomial.

Taylor Polynomial for eˣ

$$ \text{f(x)} = e^x ≈ f(x_0) + \frac{df(x_0)}{dx} (x-x_0) + \frac{d^2f(x_0)}{dx^2} \frac{(x-x_0)^2}{2!} + ... + \frac{d^nf(x_0)}{dx^n} \frac{(x-x_0)^n}{n!} $$

Therefore, when x₀= 0:

$$ \text{f(x)} = e^x ≈ e^0 + e^0(x-0) + e^0\frac{(x-0)^2}{2!} + ... + e^0\frac{(x-0)^n}{n!} = 1 + x + \frac{x^2}{2!} + ... + \frac{x^n}{n!} = P_n(x) $$

Conclusions

From the results, the determined value of e (in Python) is between $P(16) < e_{\text{python}} < P(17)$, with a maximum error ≤ $4.25 \times 10^{-16}$.

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