The Elements of the True Arithmetic of Infinites

Thus, also, in the series 1+4+7+10+13, etc. = (1+2) / (1-2+1), the number of terms is 1 / (1-1), and the last term is (1+2) / (1-1). And [1 / (1-1)] × [(1+2) / (1-1)] is equal to (1+2) / (1-2+1).

Again, in the series 1+3+6+10+15, etc. = 1 / (1-3+3-1), the last term is the series 1+2+3+4+5, etc. = 1 / (1-2+1), and the number of terms is 1 / (1-1), and [1 / (1-1)] × [1 / (1-2+1)] = 1 / (1-3+3-1).

And in the series 1+6+24+80, etc. = 1 / (1-6+12-8), the last term is (1-1) / (1-6+12-8), and the number of terms is 1 / (1-1), and [1 / (1-1)] × [(1-1) / (1-6+12-8)] = 1 / (1-6+12-8).

PROPOSITION VIII.

In every series of terms in arithmetical or geometrical progression, or in any progression in which the terms mutually exceed each other, the last term is equal to the first term, added to the second term, diminished by the first; added to the third term, diminished by the second; added to the fourth term, diminished by the third; and so on. And if the number of terms be infinite, the last term is equal to the series multiplied by 1-1.

Let the terms, whatever the series may be, be represented by a, b, c, d, e, then
a + b - a + c - b + d - c + e - d = a

• b - a
• c - b
• d - c
• e - d
  = e.

But if the number of terms be infinite, namely if the series be a+b+c+d+e+f+g, etc. to infinity, then this series multiplied by 1-1 will be