THE ELEMENTS OF THE TRUE ARITHMETIC OF INFINITES, &c.

COR. Corollary Hence, when the series is infinite, the last term of it will be equal to the series connected with a negation of itself. For by the first proposition of the 2d Book of Euclid Euclid's Elements, Book 2, Proposition 1, which states that a rectangle contained by two straight lines is equal to the sum of the rectangles contained by the first line and the segments of the second line, $a+b+c+d+e+f+g, &c.$ multiplied by $1-1$ is equal to $a \times 1-1 + b \times 1-1 + c \times 1-1 + d \times 1-1, &c. = a-a+b-b+c-c+d-d, &c.$

COR. 2. Hence, an infinite series connected with a negation of itself, is either a finite or an infinite quantity. Thus, for instance, $\frac{1}{1-1} - \frac{1}{1-1}$, is equal to $\frac{1-1}{1-1} = 1$; but $\frac{1}{1-2+1} - \frac{1}{1-2+1}$ is equal to $\frac{1-1}{1-2+1} = 1+1+1+1+1, &c.$ ad infin.

COR. 3. Hence, too, as this proposition equally applies to fractional as well as to integral consisting of whole numbers infinite series, the last terms of a great variety of fractional infinite series may be obtained, and in each of these the last term multiplied by the number of terms will be equal to the sum of the series. Thus, $\frac{1-1}{2-1} = \frac{1}{2} - \frac{1}{4} - \frac{1}{8} - \frac{1}{16},$ &c. is the last term of the series $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16}, &c. = \frac{1}{2-1}$, and $\frac{1-1}{2-1} \times \frac{1}{1-1} = \frac{1}{2-1}$. Thus, too, in the series $\frac{1}{3} + \frac{1}{9} + \frac{1}{27}, &c. = \frac{1}{3-1}$, the last term is $\frac{1-1}{3-1}$, and $\frac{1-1}{3-1} \times \frac{1}{1-1} = \frac{1}{3-1}$. In like manner in what have been erroneously called diverging series series where the sum does not approach a finite limit, but Taylor argues they have consistent fractional values, as for instance in the series $1-2+4-8+16-32, &c. = \frac{1}{1+2}$, the last term is $\frac{1-1}{1+2} = 1$