The elements of the true arithmetic of infinities

COROL. 2. Corollary 2 Hence, likewise, such expressions when they are considered as parts of infinite series, are not to be taken separate from the terms by which they are expressed, namely 1—1, for instance, is not to be considered as a subtraction of 1 from 1; for, in this case, it would be 0. Nor is 1—2 to be considered as a subtraction of 2 from 1, since it would then be —1. But these expressions are always to be considered in connexion with the numbers by which they are formed.

COROL. 3. Corollary 3 Hence, the series which are called by modern mathematicians neutral and diverging series, are erroneously so called; for they are in reality converging series.

PROP. II. Proposition 2

There cannot be a greater number of terms in any infinite series than $\frac{1}{1-1}$, which is equal to 1+1+1+1, &c. ad infinitum to infinity.

For there is an uninterrupted continuity in the series 1+1+1+1, &c. and any addition which may be made to it, does not increase the number of terms, but the quantity of some term or terms. Thus, for instance, if 1 be added to $\frac{1}{1-1}$, the sum is $\frac{1+1-1}{1-1}$, or 1+2+1+1+1+1, &c. the second term being by this mean increased, but no alteration being made in the number of terms.

PROP. III. Proposition 3

In the following infinite series, namely 1+2+3+4+5, &c. = $\frac{1}{1-2+1}$, 1+4+9+16+25+36, &c. = $\frac{1+1}{1-3+3-1}$, 1+8+27+64+125, &c. = $\frac{1+4+1}{1-4+6-4+1}$, 1+16+81+256+625, &c. = $\frac{1+11+11+1}{1-5+10-10+5-1}$, &c. and 1+32+243+1024+3125, &c. = $\frac{1+26+66+26+1}{1-6+15-20+15-6+1}$, the last term of the first series will be 1+1+1+1+1+1, &c. of the second series will be 1+3+5+7+9+11, &c. of the third series will be 1+7+19+37+61, &c. of the fourth series will be 1+15+65+175+369, &c.