# Definition: Euler's Constant

The Euler's constant $$e$$ is the value of the exponential function at the argument $$1$$:

$e:=\exp(1)=\sum_{n=0}^\infty \frac{1}{n!}=2.718~281~828~459...$

Practically, it can be calculated using the first $$N$$ terms of the exponential series $\sum_{n=0}^N \frac{x}{n&#33;}=\frac{x^N}{N&#33;}+\frac{x^{N-1}}{(N-1)&#33;}+\ldots+ x + 1$ and considering them as a real polynomial of degree $$N$$. We can then significantly reduce the number of multiplications required to calculate it by regrouping this polynomial using the Horner scheme and getting

$\sum_{n=0}^N \frac{x}{n!}=\left(\left(\ldots\left(\left(\frac xN + 1\right)\frac{x}{N-1}+1\right)\frac{x}{N-2}+\ldots\right)\frac x2 + 1 \right)+ x + 1.$

For $$x=1$$ we get the expression

$e=\sum_{n=0}^\infty \frac{1}{n!}=\left(\left(\ldots\left(\left(\frac 1N + 1\right)\frac{1}{N-1}+1\right)\frac{1}{N-2}+\ldots\right)\frac 12 + 1 \right)+ 1 + 1 + r_{N+1}$

with some remainder term $$r_{N+1}$$, which can be estimated by $r_{N+1}\le \frac 2{(N+1)!}.$

For $$N=15$$ we have $$r_{16}\le \frac 2{16!} < 10^{-13}$$ the estimation

$e=2.718~281~828~459\pm 2\cdot 10^{-13}.$

Corollaries: 1
Theorems: 2

Github: ### References

#### Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983