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Calculus Made Easy

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|- | | $1.000000$ |- | dividing by 1 || $1.000000$ |- | dividing by 2 || $0.500000$ |- | dividing by 3 || $0.166667$ |- | dividing by 4 || $0.041667$ |- | dividing by 5 || $0.008333$ |- | dividing by 6 || $0.001389$ |- | dividing by 7 || $0.000198$ |- | dividing by 8 || $0.000025$ |- | dividing by 9 || $0.000002$ |- | Total || $2.718281$ |} $\epsilon$ is incommensurable with $1$ , and resembles $\pi$ in being an interminable non-recurrent decimal.

The Exponential Series. We shall have need of yet another series.

Let us, again making use of the binomial theorem, expand the expression $\left(1+{\dfrac {1}{n}}\right)^{nx}$ , which is the same as $\epsilon ^{x}$ when we make $n$ indefinitely great.

${\begin{aligned}\epsilon ^{x}&=1^{nx}+nx{\frac {1^{nx-1}\left({\dfrac {1}{n}}\right)}{1!}}+nx(nx-1){\frac {1^{nx-2}\left({\dfrac {1}{n}}\right)^{2}}{2!}}\\&{\phantom {=1^{nx}\ }}+nx(nx-1)(nx-2){\frac {1^{nx-3}\left({\dfrac {1}{n}}\right)^{3}}{3!}}+{\text{etc}}.\\&=1+x+{\frac {1}{2!}}\cdot {\frac {n^{2}x^{2}-nx}{n^{2}}}+{\frac {1}{3!}}\cdot {\frac {n^{3}x^{3}-3n^{2}x^{2}+2nx}{n^{3}}}+{\text{etc}}.\\&=1+x+{\frac {x^{2}-{\dfrac {x}{n}}}{2!}}+{\frac {x^{3}-{\dfrac {3x^{2}}{n}}+{\dfrac {2x}{n^{2}}}}{3!}}+{\text{etc}}.\end{aligned}}$

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