The Elements of the True Arithmetic of Infinites

...and of the fifth series will be 1+31+211+781+2101, etc. The terms in each of these series are formed by the subtraction of terms from each other in the infinite series of which these latter series are the last terms. The denominators of the fractions, from the expansion of which these last terms are produced, are always one power less than the denominators of the fractions that express the sums of the series.

For 1+1+1+1+1, etc., ad infinitum, is evidently equal to the last term of the series 1+2+3+4+5, etc., ad infinitum. The sum of two of these terms, beginning from the first (1+1), is equal to the second term of the series 1+2+3+4, etc. The sum of three terms (1+1+1) is equal to 3, or the third term of the series. The sum of four terms is equal to 4, or the fourth term, and so on. Therefore, the sum of the infinite series 1+1+1+1, etc., will be equal to the last term of the series 1+2+3+4, etc. Similarly, 1+3 is equal to 4, the second term of the series 1+4+9+16, etc.; 1+3+5 is equal to 9, the third term, and so on. This holds true in the other series. It is also evident that the terms in the latter are formed from the subtraction of the terms from each other in the former series. Thus, 1+1+1+1, etc., arises from the subtraction of 1 from 2, of 2 from 3, of 3 from 4, etc.; and 1+3+5+7, etc., arises from the subtraction of 1 from 4, of 4 from 9, of 9 from 16, and so on, with the first term (1) excepted. The fractions, likewise, from the expansion of which these last terms are produced, are as follows: The fraction equivalent to the first is 1/(1-1); to the second is (1+1)/(1-2+1); to the third is (1+4+1)/(1-3+3-1); to the fourth is (1+11+11+1)/(1-4+6-4+1); and to the fifth is (1+26+66+26+1)/(1-5+10-10+5-1). The denominator of the fraction 1/(1-1) is the second power of the denominator of the fraction 1/(1-1); the denominator of (1+1)/(1-3+3-1) is the third power of 1-1; and the denominator of (1+1)/(1-2+1) is only the second power of 1-1. Thus, also, in the third series, the denominator 1-4+6-4+1 is the fourth power of 1-1, but the denominator 1-3+3-1 is only the third power of 1-1; and so on for the rest.

Corollary: If these series be supposed to begin from 0, then the expressions equivalent to their last terms will be (0+1)/(1-1), (0+1+1)/(1-2+1), (0+1+4+1)/(1-3+3-1).