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142

Calculus Made Easy

	$1.000000$
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.

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