The Elements of the True Arithmetic of Infinites

(a+b+c+d+e+f+g, etc.) / (1-1)
= (a+b+c+d+e+f+g, etc.) / (-a-b-c-d-e-f, etc.)
= a + b - a + c - b + d - c + e - d + f - e + g - f, etc.

Corollary 1. 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, a+b+c+d+e+f+g, etc. multiplied by 1-1 is equal to a × (1-1) + b × (1-1) + c × (1-1) + d × (1-1), etc. = a-a + b-b + c-c + d-d, etc.
= a - a

• b - b
• c - c
• d - d
• e - e
• f - f
• g - g,
  etc., etc.

Corollary 2. Hence, an infinite series connected with a negation of itself is either a finite or an infinite quantity. Thus, for instance, 1 / (1-1) - 1 / (1-1), is equal to (1-1) / (1-1) = 1; but 1 / (1-2+1) - 1 / (1-2+1) is equal to (1-1) / (1-2+1) = 1+1+1+1+1, etc. to infinity.

Corollary 3. Hence, too, as this proposition equally applies to fractional as well as to integral 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, (1-1) / (2-1) = 1/2 - 1/4 - 1/8 - 1/16, etc. is the last term of the series 1/2 + 1/4 + 1/8 + 1/16, etc. = 1 / (2-1), and [(1-1) / (2-1)] × [1 / (1-1)] = 1 / (2-1).
Thus, too, in the series 1/3 + 1/9 + 1/27, etc. = 1 / (3-1), the last term is (1-1) / (3-1), and [(1-1) / (3-1)] × [1 / (1-1)] = 1 / (3-1). In like manner, in what have been erroneously called diverging series, as for instance in the series 1-2+4-8+16-32, etc. = 1 / (1+2), the last term is (1-1) / (1+2) = 1.