Page:Calculus Made Easy.pdf/162

 {| style="border-style: none; margin-left: auto; margin-right: auto; width: 50%" $$\epsilon$$ is incommensurable with $$1$$, and resembles $$\pi$$ in being an interminable non-recurrent decimal.
 * $$1.000000$$
 * 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 || $$2.718281$$
 * }
 * 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 || $$2.718281$$
 * }
 * dividing by 9 || $$0.000002$$
 * Total || $$2.718281$$
 * }
 * Total || $$2.718281$$
 * }

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.